Cream the butter and sugar: In a stand mixer fitted with the paddle attachment, beat the butter and powdered sugar on medium speed until light and fluffy. Scrape down the sides of the bowl.
Add wet ingredients: Beat in the egg, vanilla extract, and peppermint extract until smooth, scraping down the bowl as needed.
Add dry ingredients: With the mixer on low, add flour and sea salt. Mix just until no visible flour remains. Use a spatula to fold in any remaining dry bits by hand.
Divide and color the dough: Split the dough in half. Remove one half and set aside. Add red food coloring to the remaining dough in the mixer and mix on low until evenly colored and vibrant.
Preheat the oven: Preheat to 375°F (190°C). Line baking sheets with parchment paper.
Shape the cookies: Unwrap both doughs. Pinch off 1-inch pieces from each color and roll into 6-inch ropes. Press one red and one white rope together at the top and twist gently to create a candy cane pattern. Curve the top to form the hook shape.
Rechill before baking: Place shaped cookies on baking sheets and refrigerate for 15 minutes before baking.
Bake: Bake for 10–12 minutes, until the edges are just beginning to turn golden. Cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely.