In a large bowl, combine the flour, sugar, baking powder, salt and ground nutmeg. In a small bowl, combine the egg, milk and butter. Add to dry ingredients and mix well.
In an pot or deep skillet, heat oil to 375°. Drop the dough by using a small round ice cream scoop, five or six at a time, into oil. Fry until browned, about 1-2 minutes, turning once.
Drain on paper towels. Roll warm doughnut holes in sugar. Serve immediately.
This recipe is adapted from Taste of Home.