Place the butter, sugar, peppermint extract, vanilla extract, baking powder, and salt into the bowl of a stand mixer. Beat these ingredients together until a paste forms and clings to the sides of the bowl.

Scrape down the sides of the bowl, then beat in the egg until fully absorbed into the mixture.

Gradually add the all-purpose flour and beat on low until just combined with no dry streaks of flour visible. Avoid overmixing the dough.

Divide the dough into two equal parts. Add the liquid red food coloring carefully to one half and mix on low speed until the red color is evenly distributed.

Shape each dough portion into a disk, wrap tightly with plastic wrap, and refrigerate for at least 3 hours or overnight to firm up and develop flavor.

Preheat the oven to 350ºF (175ºC). Line a baking sheet with parchment paper or a silicone baking mat.

Pinch off a 1-inch round piece of the white dough and roll it into a 5-inch long rope. Repeat with a red dough piece of equal length.

Place the two ropes side-by-side and twist them together gently. Transfer the twisted rope to the baking sheet and curve the top end to form a candy cane shape. Repeat with remaining dough, spacing cookies about 2 inches apart.

Beat the egg white with 1 tablespoon of water until foamy. Brush the tops of each shaped cookie with the egg white wash, then sprinkle evenly with sparkling sugar.

Bake in the preheated oven for 8 to 10 minutes or until the edges are just barely brown. Remove from the oven and allow the cookies to cool on the baking sheet for 10 minutes before transferring to a wire rack to cool completely.

Let cookie sheets cool completely before baking subsequent batches to prevent dough from spreading too much.