In a saucepan, heat the milk over medium heat just until it comes to a boil then immediately turn off the heat. Place the cornmeal in a large bowl. Add the hot milk and mix lightly. Sift together 1 cup of flour, the sugar, baking powder, cinnamon, nutmeg, and salt. Stir the melted butter and the egg into the cornmeal mixture. Stir the flour mixture into the cornmeal mixture and mix well. If the dough doesn't hold together, add more flour 1 tablespoon at a time. The dough should be firm enough to handle, but keep it soft by using as little flour as possible.
Place the dough in a bowl, cover, and refrigerate 30 minutes. Flour your hands and turn the dough out onto a lightly floured surface. Using your hands, roll into logs 1-inch in diameter. Using a knife, cut into 1 1/2-inch lengths.
Meanwhile, heat the oil in a deep fryer (or at least 2 inches of oil in a deep, heavy pot) to 350 degrees F. (Do not overheat, or the doughnuts will not cook through.) Working in batches to avoid crowding the pot, fry the dough nuts, turning occasionally, until golden brown, 4 to 5 minutes. Drain on paper towels, sprinkle lightly with confectioners' sugar, and serve immediately.