Add vegetable oil to a large, heavy Dutch oven or deep-fryer to a depth of about 3 inches. Slowly heat the oil until it registers 365 to 375 degrees on a candy or deep-fry thermometer.
Meanwhile, sift 1 cup of the flour and the salt, baking powder and baking soda into a large mixing bowl. Add the nutmeg and sugar and stir well with a wire whisk to thoroughly blend. Sift the remaining 1 1/2 cups flour into a small bowl and set aside. In a third bowl, lightly beat the egg and the egg white, then stir in the buttermilk and cooled melted butter. Add the buttermilk mixture to the dry ingredients all at once, then stir with a large looped whisk or a wooden spoon until the batter is almost smooth. Gradually beat in the remaining flour, stirring just enough to moisten the flour and smooth out most of the lumps. Do not overbeat the batter or it may become gluey and the fritters will be tough.
Have a large plate lined with several layers of paper towel near the stove. Place the cinnamon sugar in a shallow bowl and have a second platter ready to hold the finished doughnuts. When the oil has reached the right temperature, gently drop spoonfuls of the batter about the size of a large egg into the hot oil in batches of three or four. Cooking more fritters at a time may cause the oil temperature to drop too severely. Keep an eye on the thermometer--the temperature will initially jump when the batter is added, then drop slightly. Try to keep it within 5 or 10 degrees of 365 by playing with the number of fritters in the pot. Too low a heat will cause the fritters to absorb too much oil, while too high a heat will burn the outsides before the insides are properly cooked.
Fry the doughnuts for about 2 to 3 minutes on each side, turning once or twice so they cook evenly. When they are a deep amber brown, transfer them to the paper towels with a pair of tongs or a slotted spoon. Let the fritters cool for a minute or so, then roll them in the cinnamon sugar and set them on the platter (or, as in my house, directly into somebody's open, waiting hand). Allow the temperature of the oil to recover before adding the next batch of doughnuts and repeat until all of the batter is used. Like most fritters, these doughnuts are best eaten as soon as they are fried, as they tend to get soggy and unappetizing with standing.
Recipe courtesy of Regan Daley's In the Sweet Kitchen (Artisan, 2001)