Preheat your oven to 350 degrees F and grease a mini muffin tin generously with cooking spray or butter.
In a large bowl, whisk together the gluten free flour, sugar, baking powder, cinnamon, nutmeg, and salt until well combined.
In a separate bowl, whisk together the egg, milk, melted butter, and vanilla extract. Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix. The batter will be thick and slightly sticky.
Spoon the batter into the prepared mini muffin tin, filling each cavity about three-quarters full. A small cookie scoop works perfectly for this. You should get about 24 donut holes.
Bake for 11 to 13 minutes until the tops spring back when lightly touched and a toothpick inserted in the center comes out clean.
While the donut holes bake, melt the butter for the coating in a small bowl. In a separate bowl, mix together the sugar and cinnamon.
As soon as the donut holes come out of the oven, let them cool for just 2 minutes in the pan. Then pop each one out, dip it into the melted butter to coat all sides, and immediately roll it in the cinnamon sugar mixture. Place them on a wire rack.
Repeat until all donut holes are coated. Serve warm for the best flavor and texture.