1. To make the sponge cake, position a rack in the center of the oven. Preheat the oven to 325F. Brush some of the melted butter over the bottom and sides of a 10-x-15-inch rimmed baking sheet. Line the bottom of the pan and the two short sides with waxed paper. Then turn the paper over, so it is buttered side up. Brush the waxed paper again with butter. Sprinkle the waxed paper and the sides of the pan with flour, tap the pan to distribute the flour evenly, and then shake off the excess flour. 

2. In a stand mixer fitted with the whip attachment, combine 6 tablespoons of the granulated sugar, the egg yolks, and salt and beat on medium speed until light, fluffy, and very thick, about 10 minutes. Using a rubber spatula, gradually and gently fold in the flour and vanilla.

3. In a clean bowl, with the cleaned whip attachment, beat the egg whites at medium speed until big bubbles give way to tiny ones, and then start adding the remaining 3 tablespoons sugar. Turn the mixer to high speed once all the sugar is added, and continue beating until very stiff peaks form, 1 to 2 minutes longer. Using a rubber spatula, stir a “gob” of whites into the yolk mixture to lighten it, and then carefully fold in the remaining whites.

4. Pour the batter into the prepared pan, and spread it evenly to the edges using a rubber spatula or, preferably, an offset metal spatula. Bake the cake until the top is light golden brown and springs back when lightly touched, about 15 minutes.

5. While the cake is baking, spread a large linen, regular cotton, or cotton-flour-sack towel on a counter and sprinkle it generously with confectioners’ sugar. When the cake is done, remove it from the oven and immediately invert the pan onto the towel. Lift off the pan and then carefully peel the waxed paper from the cake. Cut all 4 crisp sides (edges) off the cake. Using a small fine-mesh sieve, lightly dust the top of the cake with confectioners’ sugar. Starting from a long side, roll the cake with the towel jelly-roll fashion. (The towel will be rolled inside the cake.) Let the cake cool completely in this rolled position.

6. While the cake is cooling, make the mocha buttercream. In a clean metal bowl of the stand mixer, beat together the sugar, egg whites, and salt with a wire whisk until well blended. If using a handheld mixer, whisk the ingredients together in a deep, medium heatproof bowl. Select a saucepan that will hold the mixer bowl securely in its rim, with the base of the bowl about 3 inches above the bottom of the pan. Pour water to a depth of 2 inches into the saucepan. (The base of the bowl must not touch the water.) Place the saucepan over medium heat and bring the water to a simmer. Turn the heat to low. Rest the bowl holding the sugar mixture in the rim of the saucepan and continue to whisk until the egg whites are hot and register 120F on an instant-read thermometer, about 5 minutes.

7. Remove the bowl from the water and place on the stand mixer. (If using a handheld mixer, set the bowl on a damp kitchen towel to steady it.) Using the whip attachment, beat the egg mixture on high speed until thick, glossy, stiff peaks form when you lift the beater, about 6 minutes. Add the butter 1 tablespoon at a time, beating in each addition until completely incorporated before adding the next piece. When all the butter has been incorporated the buttercream will look fluffy, satiny, and smooth. Add the coffee and mix on low speed at first and then on medium speed until the coffee is incorporated and the buttercream is again satiny smooth.

8. To assemble the cake, unroll the cooled cake with the aid of the towel. Replace the towel with a long sheet of plastic wrap and lightly sprinkle the plastic wrap with confectioners’ sugar. Lay the cake on the plastic wrap. Using a pastry brush, moisten the top of the cake with the Cognac. Set aside about 1 cup of the buttercream in a small bowl. Using an offset metal spatula, spread the remaining buttercream evenly over the top of the cake to form a thick layer, leaving a 3/4-inch-wide strip of cake without buttercream on both long sides. Starting at the long side nearest you, roll up the cake tightly and evenly (without squishing out the buttercream) into a jelly roll. If needed, use the plastic wrap to aid in the rolling. Wrap the rolled cake in plastic wrap, twisting the ends to secure them. Place the cake on a rimmed baking sheet and freeze for 1 hour.

9. Remove the cake from the freezer, transfer it to a cutting board, and unwrap it. Using a sharp knife, cut a thick, angled slice off each end of the cake. Use some of the reserved buttercream to “glue” the 2 slices together to form a stump. Use a bit more buttercream to attach the stump to the top of the cake, positioning it about 2 inches in from one end. Transfer the cake back to the baking sheet. Spread the reserved buttercream thinly and evenly over the entire log, stump, and ends. Return the cake, uncovered, to the freezer for 30 minutes. At this point, proceed to finish the cake by making the glaze and garnishes, or cover the cake tightly with plastic wrap and refrigerate for up to 2 days, or freeze for up to 2 weeks. (If you have frozen the cake, thaw the cake in the refrigerator overnight before glazing and garnishing it.) Unwrap the cake carefully before proceeding. 

10. To make the glaze, put the chocolate in a medium bowl. In a small saucepan over medium heat, bring the cream to a boil. Pour the cream over the chocolate and stir gently until the chocolate is completely melted and the glaze is shiny and smooth, about 2 minutes.

11. To finish the cake, place it on a wire rack set over a rimmed baking sheet. Pour the glaze evenly over the log, covering it completely. Use a small rubber spatula to touch up any spots that were missed along the base of the cake. Refrigerate the log on the rack for at least 1 hour to allow the glaze to set. (The cake can be glazed and refrigerated for up to 12 hours before garnishing and serving.) Scrape up the extra glaze that has dripped onto the baking sheet and save it for another use. (It can be warmed and used to drizzle over another dessert or onto dessert plates for a squiggly design. Or, it can be refrigerated until firm, rolled into balls, coated with cocoa and eaten as truffles or used to garnish the log.)

13. Make the garnishes.

For the mushrooms:

1. Position a rack in the center of the oven. Preheat the oven to 225F. Line a rimmed baking sheet with parchment paper.

2. In the bowl of a stand mixer fitted with the whip attachment, combine the egg whites and lemon juice and beat at medium-low speed until foamy. Increase the speed to medium and beat until soft peaks form, about 2 minutes. Add one-third of the sugar and beat on medium speed until combined. Add the remaining sugar and continue to beat on medium speed until combined. Increase the speed to high and beat until stiff peaks form, about 2 minutes.

3. Fit a pastry bag with a 3/8-inch plain tip. Fold back the top of the bag to form a collar. Using a rubber spatula, scoop the meringue into the bag. Twist the top of the bag closed, lightly squeezing the meringue down to the tip, forcing out any air pockets. To pipe the caps, hold the pastry bag upright with the tip slightly above the prepared baking sheet and pipe out 30 mounds, each 1 to 1 1/2 inches in diameter, on one-half of the baking sheet. They can be spaced close together because they don’t spread when baked. (I like to vary the size of the caps and stems, so the mushrooms differ in height and size when assembled.) To pipe the stems, hold the pastry bag upright with the tip touching the sheet, and squeeze and lift at the same time to form 30 cones each 3/4 to 1 inch high. Leave the remaining meringue in the pastry bag and set it aside at room temperature. Using a small fine-mesh sieve, lightly dust the caps with a little cocoa.

4. Bake the caps and stems until firm to the touch, about 45 minutes. Remove the pan from the oven. Leave the oven on.

5. Using a small paring knife, very carefully carve out a small hole in the center of the bottom of each cap. Fill the hole with a bit of the remaining meringue in the pastry bag. Attach a stem to the cap and set it upright on the baking sheet. Continue until you have made 30 mushrooms. Bake the mushrooms until set and thoroughly dry, about 30 minutes longer. Transfer the mushrooms to a wire rack and let cool completely. Store them in a single layer in an airtight container at room temperature for up to 1 month.

For the sugared cranberries and rosemary branches:

1. In a shallow bowl or pie plate, beat the egg whites just until loosened and frothy. Put the sugar in another shallow bowl or in a pie plate. Have ready a small rimmed baking sheet.

2. Dip 1 rosemary branch in the egg whites and turn to coat on all sides. Gently shake to remove any excess egg white, and then dip the branch in the sugar and turn to coat all sides. Place the sugared branch on the baking sheet. Repeat to coat the remaining rosemary.

3. Dip the cranberries in the egg whites and turn to coat all sides. Using a slotted spoon, lift the cranberries from the egg whites, allowing the excess to drip off, and transfer the cranberries to the sugar. Roll in the sugar to coat all sides. Place on the baking sheet. Set aside until ready to garnish. (Keep the plate of sugar handy to dust the rosemary or cranberries again at the last minute.)

To garnish the log, carefully transfer the cake to a long, rectangular serving plate, preferably white. Whimsically arrange some of the mushrooms around and on the log. Tuck some sugar-glazed rosemary branches and cranberries along the base of the log and near the stump, creating a woodsy scene. Dust the cake in places with a little cocoa powder to give it an earthy look. Sprinkle the plate and cake with confectioners’ sugar to imitate freshly fallen snow. Present the cake whole and slice it at the dining table, or place it whole on the buffet table.

Diane Morgan, The Christmas Table, Chronicle Books (2008).