IN A LARGE BOWL, whisk together 1/2 cup of the sugar with the salt, baking powder, and nutmeg. Add the milk, egg, melted butter, and vanilla and use a wooden spoon to mix until incorporated. Add the flour and mix well until the dough is uniform and smooth. Cover the dough with plastic wrap and refrigerate for 1 hour. In a deep fryer or large deep skillet, bring a few inches of oil to 375°F.
Dust a baking sheet generously with flour and set aside. Add the remaining 1 cup of sugar to a medium bowl and set aside. Turn out the dough onto a well-floured surface and use a rolling pin to roll it out to about a 1/2 inch thickness. Use a 2 1/2-inch doughnut cutter to cut out doughnuts, dipping the cutter in flour as needed to prevent sticking. Place the doughnuts and doughnut holes on the floured baking sheet 2 inches apart. Re-roll the scraps, cut them out, and repeat until you've used all of the dough.
Working in batches to avoid overcrowding, gently drop the doughnuts into the hot oil (the oil temperature will drop too much if you overcrowd the fryer, and you'll end up with soggy, greasy doughnuts). Fry on the first side for about 1 minute, until golden brown, then flip and cook for another minute or so, until golden brown on the second side. Transfer the fried doughnuts to the bowl of sugar and toss to coat, then arrange on a serving platter and grate fresh nutmeg over the top. Serve.
