In a large mixing bowl or stand mixer beat the butter, cream cheese, sugar, and egg until nice and frothy. About 4 minutes. Add the vanilla and peppermint extract and beat for 1 minute more.
In a small bowl combine the flour, baking powder and baking soda. Mix them around so that all ingredients are well incorporated.
Add the flour mixture to the butter mixture and beat for about 2 minutes.
Wrap dough in plastic wrap and chill for 1 hour. Dough will be a little sticky.
Preheat oven to 350 F and line insulated baking sheets with parchment paper or silicone baking mats.
Portion out dough balls with a cookie scoop or by heaping tablespoons and place 2-inches apart on prepared baking sheets.
Bake 12-14 minutes, edges should be very lightly browned.
Cool on pan ten minutes then move to racks and allow to cool completely.
To Make the Frosting
In a medium-sized bowl (or your stand mixer) combine the butter and cream cheese. Beat for about 2 minutes.
Add the powdered sugar, milk, vanilla extract, peppermint extract, and red food coloring. Mix for 1-2 minutes or until smooth.
Decorate the Cookies
When cookies are cool and you are ready to decorate, unwrap the candy canes and place in a zipper plastic bag. Crush with a meat mallet.
Spread 3-4 cookies with frosting and sprinkle with crushed candy canes. Repeat until all cookies are decorated.
Notes
Decorating a few at a time helps the candy cane stick to the frosting before it has a chance to dry.
If you would like to make these cookies a little larger, simply bake for a few minutes more.
The calories listed may be overstated if you don't end up using all the frosting.