In a large bowl, using a hand mixer or in the bowl of a stand mixer with a paddle attachment, cream together the butter and the powdered sugar at medium speed until fluffy.
Add the egg, vanilla and peppermint extract, and salt. Mix until well combined.
Add the flour and mix until well combined.
Divide the cookie dough in half and stir the red food coloring into ½ of the dough portion. Wrap in plastic wrap and chill for up to 30 minutes.
Remove the dough from the refridgerator and preheat the oven to 375 degrees.
Divide the red dough and the white dough into 1 teaspoon portion sizes. Roll each portion of dough into a 4" rope.
Place 1 white and 1 red rope side by side and twist them together, pinching the ends to seal. Form the twisted rope into a candy cane. Place the cookies 2" apart on a baking sheet that has been lined with parchment paper.
Bake for 8 minutes or until cookies are firm to the touch. Remove and cool on a wire rack.
Store the cookies in an airtight container for up to 1 week on the counter and 3 months in the freezer.