1.Combine 1/2 cup of the sugar and 1/4 teaspoon of the cinnamon in a small paper bag and shake until thoroughly combined. Set aside.

2. In a large bowl, add the flour, baking powder, baking soda, salt and the remaining 1/4 teaspoon of cinnamon and mix to combine. Set aside.

3. In the bowl of a stand mixer on medium-high speed, beat the egg until foamy. Gradually add the remaining 1/2 cup of sugar and beat until the mixture is thick and yellow, about 1 minute.

4. Meanwhile, in a small pan over low heat, combine the butter and milk, cooking just until the butter has melted. On low speed, add the warm milk-butter mixture to the egg-sugar mixture, then the vanilla extract and beat to combine.

5. Using a spatula, add the wet ingredients to the dry ingredients, mixing until just incorporated; a few lumps may remain. Cover and refrigerate for at least 1 hour until the dough is cold.

6. On a lightly floured work surface, gently pat out the dough to a thickness of about 1/4 inch (it will be somewhat loose and wet). Use a 3- to 4-inch doughnut cutter or 2 circular biscuit cutters that are in 3-inch and 2-inch sizes to make a first round of doughnuts; pat out the dough only 1 more time to keep it from being overworked or too floury. (Dip the cutters' edges in flour between cuts.) Let the formed doughnuts rest on a rimmed baking sheet at room temperature for a few minutes or until they form a slight crust. At this point, the doughnuts may be wrapped and frozen until ready to cook.

7. In a heavy-bottomed, deep pot, heat 4 inches of oil to 375F on a candy thermometer. Add a few doughnuts at a time and cook until they are light brown on the underside, then flip over and continue to cook, 1 to 2 minutes total. (Keep an eye on the thermometer and adjust as necessary to keep the temperature steady.) Transfer briefly to blot on paper towels, if desired, then add a few doughnuts at a time to the bag containing the cinnamon-sugar mixture and shake to coat evenly. Serve warm.