Fill a large heavy-bottomed pot with oil, about 3-4 inches deep. Place over medium heat until a deep-fry thermometer registers 335°F. Line a large baking sheet with paper towels.
In a medium bowl, whisk together the gluten-free flour, baking powder, salt and nutmeg until well-combined.
In a separate bowl, whisk together the granulated sugar, milk, oil and egg until smooth.
Add the wet ingredients to the dry and mix with a rubber spatula to evenly combine.
When the oil is ready, scoop 1 tablespoon-sized holes of donut batter into the hot oil. Cook for 2 minutes, then turn over and cook for an additional 2 minutes or until they are deep brown.
Remove the donut holes from the oil and let them drain on the baking sheet. Repeat with the remaining batter.
Mix the granulated sugar and cinnamon in a small bowl. Scoop the powdered sugar into another small bowl.
Once the donut holes are cool enough to handle, coat in either sugar.