In an 8-inch ovenproof skillet, melt the butter. Scrape the omelet mixture into the skillet and shake gently to evenly distribute the eggs. Sprinkle the cheese all over and transfer the skillet to the oven. Broil the omelet for about 3 minutes, until lightly browned and very puffy. Carefully slide the omelet onto a plate, folding it in half. Serve right away.