Combine the whole eggs, egg whites, and water in a medium bowl and whisk well.
In a medium ovenproof nonstick skillet, heat the oil over medium heat. Add the onion and cook, stirring until it begins to soften, about 3 minutes. Add the broccoli and cook for another 2 minutes. Season with salt and pepper. Pour the egg mixture over the vegetables in the skillet, covering them evenly. Reduce the heat to medium-low, cover, and let cook until the egg mixture has set around the edges of the pan but is still somewhat liquid in the middle, about 8 minutes. Sprinkle with the cheese.
Place the skillet under the broiler about 2 inches from the heat until the surface is set and golden brown, 1 to 2 minutes. Be careful not to overcook or the egg mixture will become tough.