In a large mixing bowl, cream the softened butter and powdered sugar until light and fluffy. Add in the egg, peppermint extract, and vanilla extract. Mix until well combined.
In a separate bowl, whisk together flour and salt, then gradually add to the wet mixture. Once a soft dough forms, divide it into two equal parts. Leave one part plain and add red food coloring to the other. Mix until the red dough is evenly colored. Wrap both portions in plastic wrap and chill for 30 minutes.
Preheat the oven to 350°F (175°C). Take a small piece (about 1 teaspoon) of each dough color. Roll each piece into a rope about 4 inches long. Twist one red and one white rope together and curve the top to form a candy cane shape. Place on a parchment-lined baking sheet.
Bake for 8–10 minutes, or until the edges are just beginning to turn golden. Do not overbake. Allow to cool on the pan for a few minutes before transferring to a wire rack. Optional: Sprinkle crushed candy canes on top while warm.