Preheat your oven to 350°F (175°C) and lightly grease your mini muffin pan.
In a large mixing bowl, combine the dry ingredients: gluten-free flour, sugar, baking powder, baking soda, salt, cinnamon, and nutmeg.
In another bowl, mix the wet ingredients: buttermilk, egg, melted butter, and vanilla extract until smooth.
Pour the wet mixture into the dry ingredients and stir until just combined. Spoon the batter into the prepared mini muffin pan, filling each cup about 3/4 full.
Bake for 10-12 minutes until golden brown and a toothpick comes out clean. Let cool in the pan for 5 minutes.
Prepare the glaze by whisking together powdered sugar, milk, and vanilla extract until smooth.
Dip the tops of each donut hole into the glaze and place them on a wire rack to let the glaze set.