Preheat the oven to 350°F. Use some extra softened butter to GENEROUSLY grease the donut wells in a donut pan.
In a medium bowl, add the rice flour, brown sugar, cocoa powder, baking powder, and salt. Whisk well to combine, especially taking care to break up any brown sugar lumps.
In a large bowl, add the melted butter, coconut milk, eggs, and vanilla. Whisk well to combine until smooth, this may take a bit as the coconut milk can come out chunky from the can. Stream in the melted chocolate while whisking, and continue to whisk until everything is combined.
Add half the dry ingredients into the wet and whisk until smooth, and then add the rest of the dry ingredients and continue mixing until smooth. I recommend adding the batter to a large piping bag (or ziploc if you don't have piping bags) and pipe into your donut pan, filling the wells pretty much right up to the top. Smooth out the batter if necessary with a small spatula.
Bake the donuts for 30-35 minutes. They should be set on top and gently spring back when poked. You wan't them to feel a bit underdone, as they will set firmer once cooled.
Let them cool for 10 minutes in the pan, and then turn out onto a wire rack to cool completely before glazing. If they get a bit stuck (which I find happens with my donut pans) gently use a knife or small offset spatula to free the donut sides and then tip out onto a wire rack.