In a medium bowl, whisk together flour, cocoa powders, baking soda, and salt. Set aside.
In the bowl of a stand mixer, beat butter and sugars on med-high until light and fluffy (approx. 2-3mins). Reduce speed and add egg and peppermint extract. Beat until well combined. Add flour mixture and mix until just combined.
Chill dough for at least 1 hour or overnight.
Preheat oven to 350°F and line baking sheets with parchment paper or silicone mats.
Using a medium cookie scoop, portion dough onto baking sheet (I did 9 cookies per sheet). Bake for approximately 10 mins, or until the middle is set. Cookies will deflate and crinkle a bit upon cooling.
Cool on pans for 5 mins then transfer to to a wire rack to cool completely.
Dip in melted candy melts or white chocolate, place on a parchment lined baking sheet and sprinkle with crushed candy canes. Allow to set.
* Read this post for details on how to best melt candy melts.