In the bowl of a stand mixer fitted with the paddle attachment, mix together the butter and powdered sugar on medium speed until creamy. Add the egg, almond extract and vanilla extract and mix until combined.
In a medium bowl, whisk together the flour, baking powder and salt. With the mixer on low speed, mix in the flour mixture until just combined.
Divide the dough in half (leaving one half of the dough in the mixing bowl). Mix red food coloring into the half of the dough in the mixing bowl; leave the other half of the dough white. (If you want half of your cookies to be red and white, and half to be green and white, tint ¼ of the dough red and ¼ green, and leave ½ white.)
Shape each half of the dough into a disk and wrap in plastic wrap. Refrigerate for at least 2 hours.
While the dough chills, make the peppermint sugar topping: In a small bowl, mix the crushed peppermints and granulated sugar together. Set aside.
Heat oven to 350° F. Line baking sheets with parchment paper.
To make each cookie, on a very lightly floured work surface, shape 1 heaping teaspoon of white dough and 1 heaping teaspoon of red dough into smooth balls and then roll each ball into a 5-inch rope. Lay the red and white ropes side by side and gently twist them together. Shape the twisted ropes into candy canes and place on parchment-lined baking sheets.
Bake cookies for 8-10 minutes, until just set and barely golden on the bottom.
As soon as the cookies come out of the oven, sprinkle them with the peppermint-sugar mixture. Let cookies cool for a few minutes on baking sheet before transferring to a wire rack to cool completely.