Place the softened butter and powdered sugar in a large bowl or stand mixer bowl. Beat on medium speed for 2 to 3 minutes until light and fluffy. Add the egg, vanilla, and peppermint extracts and mix well to combine.

Gradually incorporate the all-purpose flour, adding one cup at a time and mixing on low speed to combine completely between additions. Scrape down the bowl sides and bottom as needed for even mixing.

Remove half of the dough and form it into a one-inch-thick disc. Wrap tightly with plastic wrap and refrigerate for at least 1 hour or overnight to firm up.

Add red food coloring to the remaining dough and mix on low until fully incorporated. Knead on a clean surface to distribute the color evenly, form into a one-inch-thick disc, wrap with plastic wrap, and chill for 1 hour.

Preheat your oven to 375°F (190°C). Line two large baking sheets with parchment paper.

Unwrap both dough discs. Pinch off 1/2-tablespoon pieces from each color separately. Roll each piece into a 5-inch rope, keeping red and white doughs separate. Place a red rope next to a white rope and twist them together, pinching ends to seal. Shape the twisted ropes into candy cane forms and place on baking sheets. Refrigerate dough if it becomes too sticky.

Bake the cookies for 12 minutes or until the white dough is just lightly golden. Immediately sprinkle crushed candy canes over the cookies if using. Let cookies cool completely on the baking sheets.

Once cooled, store the cookies in an airtight container in a single layer separated by parchment paper. They keep fresh for up to 1 week at room temperature.