Preheat the oven to 350°F (175°C) and line a large baking sheet with parchment paper.
In a medium bowl, whisk together the gluten-free flour, cocoa powder, baking soda, and salt until no lumps remain.
In a separate large bowl or stand mixer, beat the softened butter with the brown sugar and white sugar until light and fluffy.
Add the egg, peppermint extract, and vanilla extract to the butter mixture and beat until well combined.
Gradually stir the dry ingredients into the wet ingredients until a thick dough forms.
Fold in the chocolate chips and the crushed peppermint candy cane pieces.
Scoop the dough into 1.5-inch balls (approximately 1 tablespoon each) and place them 2 inches apart on the prepared baking sheet.
Bake for 8 to 10 minutes until the edges are set but the centers still look slightly soft.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.