In a mixing bowl, beat together the butter, shortening, and sugar with an electric mixer until light and fluffy; about 3 minutes.
Scrape down the sides of the bowl and add the egg, vanilla extract, and peppermint extract. Continue to mix until well incorporated.
Add the flour and salt; mix together on low speed until just combined.
Remove dough from mixing bowl and divide the dough in half; place each half into its own bowl.
Add the red food coloring to one of the bowls and mix together well. You can add extra food coloring if you prefer a brighter shade.
Form both pieces of dough into 1-inch-thick disks and refrigerate for at least an hour.
When ready to bake, preheat the oven to 375˚F and line a baking sheet with parchment paper.
Unwrap the disks of dough and pinch off 2-tablespoons piece of dough from each disk. Roll each piece of dough into a thick, 6-inch long rope.
Pinch the red and the white rope together at the top, and gently twist and braid them together. Form a bend at the top of the twisted rope to make a candy cane shape; then, transfer to the prepared baking sheet. Repeat this process with the remaining dough.
Refrigerate the cookies for 20 minutes before baking.
Bake for 10 to 12 minutes and allow the cookies to cool on the baking sheet for 10 minutes before transferring to a wire cooling rack.
Remove from rack and serve.