Preheat the oven to 325ᴼF. Grease an 8-inch square baking dish with oil and line the bottom with a piece of parchment paper.

Set aside.

In a medium bowl, whisk together the white and brown rice flours, tapioca starch, baking powder, and salt.

In the bowl of an electric mixer fitted with the whisk attachment or using a handheld mixer, beat the coconut oil and sugar on medium speed until well combined, about 2 minutes. Add the cocoa powder, vanilla, and peppermint extract and beat until incorporated. Add the eggs one at a time, beating well after each addition. Add the flour mixture ½ cup at a time, beating until just combined. Stir in the walnuts and spread the batter in the prepared pan.

Bake until a toothpick inserted into the center comes out nearly clean, about 40 minutes. Let cool to room temperature before cutting and serving.