Heat oven to 375 degrees F.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and powdered sugar together. Add the egg and extracts and mix until combined.
In a medium bowl, whisk together the flour and salt. With the mixer on low speed, mix in the flour and salt until just combined.
Divide the dough in half. Add red food coloring to one half and leave the other half 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.)
Wrap dough in plastic wrap and 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.
To make each cookie, shape 1 rounded teaspoon of white dough and 1 rounded teaspoon of red (or green) dough into 4-inch ropes. Lay the 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 9 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 cool for a few minutes on baking sheet before transferring to a wire rack to cool completely.