Place the butter, sugar, peppermint extract, vanilla extract, baking powder, and salt in the bowl of a stand mixer. Beat until a paste forms and clings to the side of the bowl. Scrape down the side of the bowl and beat in the egg until it is absorbed.

Add the flour and beat until just combined and no more dry streaks of flour remain.

Divide and remove half the dough from the bowl. Carefully add the red food coloring to the remaining dough. Mix on low until the color is evenly distributed.

Pat each piece of dough into a disk, wrap in plastic wrap, and refrigerate for at least 3 hours or overnight.

Preheat the oven to 350ºF. Line a baking sheet with parchment paper or a silicon baking mat.

Pinch off a 1-inch round piece of the white dough and roll it out to a 5-inch rope. Do the same with a piece of the red dough, ensuring the ropes are the same length. Place the ropes next to each other and twist together. Curve the top of the dough into a candy cane shape and transfer to the baking sheet, spacing the cookies about 2 inches apart.

Make the egg white wash by beating the egg white with 1 tablespoon of water until foamy. Brush the top of each cookie with the egg white wash and then sprinkle with the sugar.

Bake in the oven for 8 to 10 minutes or until the edges are just barely brown. Let cool on the baking sheet for 10 minutes before moving to a wire cooling rack.