Preheat oven to 350 degrees F.
Melt the butter in a 7-inch nonstick oven-proof skillet over medium-high heat. Cook the onions and peppers until beginning to soften, about 2 minutes. Add the green onions and mushrooms and cook until soft, about another 2 minutes. Season with salt and pepper.
Whisk the eggs in a bowl until well blended and season them with salt and pepper. Pour the eggs over the vegetables in the skillet. Stir constantly with a heat-proof rubber spatula. When the eggs are almost cooked, spread them evenly over the pan. Cover the pan and put it into the oven for 1 minute. Remove the cover and sprinkle the cheese evenly over the eggs. Return the skillet to the oven and cook until the cheese is melted, about 2 minutes.
Note: Any vegetables can be used. Meat can also be added if it is cooked first.