Adjust an oven rack to the middle position and preheat the oven to 350ºF. Grease two 6-cavity doughnut pans with nonstick cooking spray, melted coconut oil, or melted butter.

Whisk the coconut sugar, coconut flour, tapioca starch, baking powder, nutmeg, and salt together in a medium mixing bowl. Add the melted coconut oil, eggs, milk, and vanilla and whisk until smooth. The batter will be thick.

Spoon the batter into the prepared pans, filling the cavities about three-quarters full. (If you have a piping bag and large plain pastry tip kicking around the kitchen, use those instead.) Lightly tap the pans on the counter to settle the batter.

Bake the doughnuts until they are golden brown and spring back to the touch, about 15 minutes.

Allow the doughnuts to cool in the pans on a wire rack for 5 minutes, then turn them out onto the rack to cool completely.

Doughnuts are best enjoyed the day they are baked. Freeze leftovers, wrapped in plastic wrap and placed in a freezer container, for up to one month.