Preheat the oven to 350 degrees and line 2 baking sheets with parchment paper.
In a small bowl, combine the gluten free flour, cocoa powder, baking soda and salt. Whisk until well mixed. Set aside.
Add butter and sugar to the bowl of an electric mixer. Beat on medium high speed for 2-3 minutes until light and fluffy. Scrape down the bowl.
Add the egg and vanilla. Beat another 60 seconds on medium speed. Scrape down the bowl.
Turn the mixer to the lowest setting and slowly add the flour mixture. Stir until just combined. Use a spatula to ensure all dry ingredients have been incorporated into the batter.
Scoop the dough into balls the size of grapes and place on your prepared baking sheets spaced about 2 inches apart.
Bake for 7 minutes in preheated oven until edges are just set. Cool for 5 minutes on the baking sheet, then move to a wire rack to cool completely. The cookies will be soft.
For the frosting
Add butter to the bowl of an electric mixer. Beat on medium speed about 1 minute until the butter is smooth. Scrape down the bowl.
Add the sifted powdered sugar and peppermint extract to the butter. Beat on medium low until the dry ingredients are incorporated and it looks a little like a dough ball. Scrape down the bowl to make sure all of the ingredients are incorporated evenly. Then turn your mixer to high speed and beat for 5 minutes. The frosting should be light and fluffy.
Turn half of your cookies over so the bottoms are facing up. Scoop about 1 ½ tablespoon of frosting onto each cookie half. You can use a frosting bag if you want, but I just use a spoon. Place the second cookie half on top of the frosting and gently press down so the frosting spreads to the edges.
Place your candy canes (or peppermint candies) into a freezer safe ziploc bag and seal. Smash using a wooden rolling pin or something equally as heavy and strong.
Roll the edges of the cookies in the crushed candy canes.
Store cookies in an airtight container.
Notes
If you wish to substitute another brand of gluten free flour, I recommend using a kitchen scale to weight it to 166g. For King Arthur's Measure for Measure this is approximately 1 ⅓C