In the bowl of a stand mixer (or with a hand mixer), beat butter and confectioners' sugar on medium speed until the batter looks soft, about 90 seconds. Scrape down the sides of the bowl, then add the egg, vanilla, and peppermint. Mix until combined. With the mixer on low speed, slowly add in the flour and salt. Mix just until combined (avoid overmixing).
Remove half the dough from the mixing bowl and set aside. Add 1/2 teaspoon of red food coloring to the remaining dough and mix on low speed. Add additional food coloring as needed until the dough is a bright, vibrant red.
Form each doughs into 1-inch thick discs, wrap each separately in plastic, and refrigerate for at least 1 hour or up to 3 days.
Preheat the oven to 375° F. Line two baking sheets with parchment paper and set aside.
Remove both pieces of dough from the fridge and unwrap. Pinch off roughly 2 tablespoon and roll each into a thick rope about 6 inches long. Pinch the two ropes together at the top and gently twist them together. Transfer the twisted ropes to the baking sheet and gently "hook" the top to create a candy cane shape.
Repeat with the remaining dough. If the dough starts to break as you roll it or become hard to work with, put it back in the fridge to chill for 15 minutes. When all the cookies have been shaped, chill both sheets of cookies for 15 minutes before baking — this helps the cookies hold their shape during baking.
Bake for 10 to 12 minutes, until the edges and tips of the cookies are just barely golden. Cool for 10 minutes on the baking sheet and then transfer to a cooling rack to cool completely.
Store cookies in an airtight container for up to a week.