In a large bowl, combine the flour, sugar, baking powder, salt and nutmeg. In a small bowl, combine the egg, milk and butter. Add to dry ingredients and mix well.
In an electric skillet or deep-fat fryer, heat oil to 375°. Drop dough by heaping teaspoonfuls, 5 or 6 at a time, into oil. Fry until browned, about 1-2 minutes, turning once. Drain on paper towels. Roll warm doughnut holes in confectioners' sugar.