Heat the oil in a medium nonstick ovenproof skillet. Add the chopped onions and cook over moderately high heat, stirring, until sizzling, about 2 minutes. Reduce the heat to moderately low and cook, stirring frequently, until the onions are well browned, about 30 minutes longer.
Meanwhile, in a skillet of boiling salted water, cook the asparagus until tender, 3 to 4 minutes; drain and set aside.
Preheat the broiler. In a bowl, whisk the whole eggs, egg whites and 1 tablespoon of water until frothy, then season with salt and pepper. Pour the eggs over the hot onions in the skillet and let set for 1 minute. Cover and cook until the eggs are set around the edges but still soft in the center, 2 to 3 minutes.
Arrange the asparagus on top of the frittata, then slip the pan under the broiler and cook for 30 to 60 seconds, or until the omelet puffs up and lightly browns. Slide the frittata onto a serving plate and sprinkle with the chopped parsley. Cut the frittata into 6 wedges and serve.
One Serving Calories 100 kcal, Protein 8 gm, Carbohydrates 8 gm, Cholesterol 142 mg, Total Fat 4.3 gm, Saturated Fat 1.2 gm.