Preheat oven to 350℉. Sift the gluten free flour, cocoa powder, baking powder, and salt. Set aside. Whisk the eggs, oil, sugar, milk, and vanilla extract together. Add the dry ingredients and mix until combined.
Pour the batter into a greased donut pan and bake for 8-12 minutes, or until a toothpick comes out clean. Let the donuts cool completely.
Whisk the icing ingredients until smooth. Dip the cooled donuts in the icing and then coat in rainbow sprinkles. Let the icing harden before serving.