Preheat the oven to 350°F (175°C) and line two large baking sheets with parchment paper.

Whisk together the gluten-free flour, cocoa powder, baking soda, and sea salt in a medium bowl until uniform.

Cream the softened butter, brown sugar, and granulated sugar in a stand mixer on medium-high speed for 3 minutes until light and aerated.

Add the egg, vanilla extract, and peppermint extract to the creamed mixture and beat until fully emulsified.

Reduce mixer speed to low and gradually incorporate the dry ingredient mixture until no dry streaks remain.

Fold the chocolate chips into the dough using a silicone spatula.

Refrigerate the dough for 30 minutes to allow the gluten-free flours to hydrate and to prevent excessive spreading during baking.

Portion the dough into 1.5-tablespoon spheres and place them 2 inches apart on the prepared sheets.

Bake for 10 to 11 minutes until the edges are set and the surface shows slight cracking, but the center remains soft.

Press crushed candy canes into the surface of the warm cookies immediately upon removal from the oven.

Cool the cookies on the baking sheet for 10 minutes to allow the internal structure to set before transferring to a wire rack.