In a large mixing bowl, whisk together the all-purpose flour, 1/2 cup of granulated sugar, baking powder, salt, ground cinnamon, and nutmeg until thoroughly combined.
In a separate medium bowl, mix together the milk, beaten egg, and melted butter until smooth.
Gradually pour the wet ingredients into the dry ingredients, stirring gently until just combined.
In a deep fryer or heavy-bottomed pan, heat oil to 350°F (175°C).
Using a small cookie scoop or your hands, portion out about 1 tablespoon of batter and drop it into the hot oil.
Fry the donut holes for 2-3 minutes on each side, or until golden brown.
Using a slotted spoon, remove the donut holes from the oil and transfer them to a paper towel-lined plate.
In a medium bowl, mix together the 1/2 cup of granulated sugar and 1 tablespoon of ground cinnamon.
While warm, roll each donut hole in the cinnamon sugar mixture until coated.