Preheat the oven to 350°F. Line large baking sheets with parchment paper or silicone baking mats.
In a medium bowl, whisk together the gluten-free flour, cornstarch, baking powder and salt.
In the bowl of a stand mixer fitted with a paddle attachment, or using a hand mixer, add the butter and powdered sugar and mix on medium speed for 2-3 minutes, until light and fluffy.
Add the egg, peppermint extract and vanilla extract and mix until fully incorporated. Scrape down the sides and bottom of the bowl.
With the mixer running on low, slowly add the dry ingredients to the wet ingredients and mix until well-combined. Scrape down the sides and bottom of the bowl, then mix again until fully incorporated.
Scoop the cookie dough into round balls, 1 tablespoon per ball. Roll the balls until smooth. Place on the prepared baking sheet, about 2 inches apart.
Bake for 11-13 minutes, until the cookies are set.
Remove from the oven and cool on the baking sheets for 10 minutes. Transfer to a wire rack to cool completely.
For the peppermint glaze: Melt the butter and coconut oil in a small saucepan or microwave-safe bowl. Whisk in the powdered sugar, milk and peppermint extract until smooth.
Use a toothpick to add the tiniest amount of red food coloring to the icing. Stir and repeat as necessary until you reach your desired color.
Dip, brush or spread the warm glaze onto the cooled cookies. Microwave the glaze for about 10 seconds to slightly thin it if it gets too thick. Immediately sprinkle with crushed peppermint candies before the glaze sets.