In a large heavy skillet, combine onion and olive oil; slowly cook over low heat until onion is soft and golden brown. Drain off oil.
In a medium bowl, combine eggs with about 1 to 2 tablespoons water and salt and white pepper to taste, if desired; with a wire whisk, vigorously whisk egg mixture into a slight froth. Add the onion and Parmesan cheese, mixing well.
Melt butter in a 12-inch diameter heavy skillet or omelet pan over low to moderate heat. Add egg mixture, coating the bottom of the skillet. With a fork, gently scramble the egg mixture in a circular motion until partially set.
After several minutes of very slow cooking, when the eggs have set and thickened, and only the surface is runny, place the skillet under the broiler to lightly brown omelet, about 1 minute or less.
To serve, loosen the frittata with a spatula and slide it onto a platter. Cut it into wedges and serve immediately.
Variations: Reduce onions to 3 cups and add about 1 cup cooked peeled deveined very small or coarsely chopped shrimp to the frittata/omelet with the onion/cheese mixture. Reduce onions to 3 cups add about 1 cup crumbled sautéed Italian sausage or coarsely chopped proscuitto to the frittata/omelet with the onion/cheese mixture.