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 dough into 1-inch thick discs, wrap each separately in plastic, and refrigerate for at least 1 hour or up to 24 hours days.
Line two baking sheets with parchment paper and set aside. Remove both pieces of dough from the fridge and unwrap. Scoop off roughly 1 1/2 tablespoons of white dough and 1 1/2 tablespoons of red dough. 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.
When all the cookies have been shaped, chill both baking sheets for 15 minutes. This helps the cookies hold their shape during baking.
While the cookies are chilling, preheat the oven to 375° F.
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.
Notes
Flavoring: If you prefer almond flavor in your shortbread, you can omit the peppermint extract and use 1/2 teaspoon almond extract instead.Crumbly dough breaking: If the dough starts to break as you roll it or it becomes hard to work with, put it back in the fridge to chill for 15 minutes. If that doesn't work, add 1-2 tablespoons milk to the cookie dough. It's likely there was too much flour added to the dough and it was too dry. Store cookies in an airtight container at room temperature for up to a week.Freeze dough wrapped in plastic wrap for up to 3 months. Defrost in the refrigerator before shaping cookies.