Separate 4 of the eggs, putting the whites into a medium sized bowl and discarding the yolks. Add the 4 whole eggs and 2 tablespoons of water to the whites and whisk well.
In a medium ovenproof nonstick skillet heat the oil over a medium flame. Add the onion and cook until it begins to soften, about 5 minutes. Add the broccoli and cook for another 2 minutes. Season with salt and a few turns of 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 but is somewhat liquid in the middle, about 8 minutes. Sprinkle with the cheese.
Meanwhile, preheat the broiler. 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.