Preheat the oven to 350°F. Line baking sheets with parchment paper.
In a medium bowl, whisk together the gluten-free flour, cornstarch, baking powder, and salt.
In a stand mixer, cream butter and powdered sugar for 2-3 minutes until light and fluffy.
Add the egg, peppermint extract, and vanilla extract; mix until fully incorporated.
Slowly add dry ingredients to wet ingredients and mix until well-combined.
Scoop 1 tablespoon of dough per ball, roll smooth, and place on the prepared baking sheet 2 inches apart.
Bake for 11-13 minutes, until the cookies are set.
Cool on baking sheets for 10 minutes, then transfer to a wire rack to cool completely.