Preheat the oven to 350 F. Prepare 2 standard cupcake pans by lining with cupcake liners.
In a mixing bowl, mix all dry ingredients on low speed to blend. Add the soft butter and mix until evenly crumbly, like fine damp sand.
Add the egg whites one at a time, then the whole egg, beating well after each addition to begin building the structure of the cake. Scrape down the sides and bottoms of the bowl after each addition/
In a small bowl, whisk the milk with the vanilla extract. Add this mixture, 1/3 at a time, to the batter. Beat 1 to 2 minutes after each addition, until fluffy. Be sure to scrape down the sides and bottom after each addition in the bowl.
Pour the batter into the prepared pans. Bake for 20 minutes. A toothpick or cake tester inserted in the center will come out clean when done. Remove from oven, remove from pan, and cool on a rack. Frost with white chocolate peppermint buttercream.
In a large mixing bowl, using a handheld electric mixer at medium speed, cream butter until light and creamy. Beat in the melted white chocolate.
Add the confectioners' sugar and peppermint extract and beat a low speed, scraping sides and bottom of bowl, until light and fluffy.
I made my own superfine sugar using Dixie Crystals.
Cupcake recipe adapted from King Arthur Flour