Preheat oven to 350 degrees. Spray two doughnut pans with cooking spray.
In a large mixing bowl, whisk together the brown rice flour, ½ cup tapioca starch, baking powder, and salt. In another mixing bowl, whisk the eggs with the coconut oil, 3/4 cup coconut sugar, 1 cup coconut milk and the vanilla extract. Add the wet ingredients to the dry ingredients and mix together. Spoon the batter into the prepared doughnut pans and smooth the tops of the doughnuts, making sure the post in the middle is free of batter.
Place the coconut flakes on a baking sheet, spread in an even layer. Put the filled doughnut pans and the baking sheet with the coconut flakes in the preheated oven. Cook the coconut for 5 – 6 minutes or until browned and fragrant. Remove the coconut from the oven and continue to bake the doughnuts for another 6 or 8 minutes (12 -13 minutes total) or until golden brown and the doughnuts feel firm to the touch. Let the doughnuts cool in the pans for a few minutes, then remove to a wire rack set over a piece of parchment or waxed paper to finish cooling.
While the doughnuts are cooling, prepare the glaze. Place the remaining 1 cup of coconut sugar in a high powered blended or spice grinder with the remaining 2 teaspoons of tapioca starch and process until it turns into a powder. If using a spice grinder, you will need to do this in 2 batches – ½ cup coconut sugar with 1 teaspoon tapioca starch per batch. Pour the mixture into a small bowl, stir with a fork to break up any lumps and add the remaining coconut milk, 1 tablespoon at a time, until a thick glaze is achieved. You may need a little more or less than 4 tablespoons.
Crush the coconut flakes with your hand into smaller pieces and place on a plate. Frost each doughnut with some of the glaze, then dip the glazed side into the coconut flakes. Set on cooling rack and allow the glaze to dry. Repeat with remaining doughnuts.