Heat the oven to 350°F or turn on the broiler. Put 1 tablespoon of olive oil in a large skillet over medium-high heat. When it’s hot, add the onion and cook, sprinkling with salt and pepper, until soft, about 5 minutes.
Meanwhile, beat the eggs in a bowl with some salt and pepper until the yolks and whites are just combined. Pour the eggs over the onions, using a spoon if necessary to evenly distribute them. Sprinkle the cheese over the top and cook, undisturbed, until the eggs are barely set, 5 to 10 minutes.
Transfer the skillet to the oven and cook until the edges brown and puff but the center is still jiggly, about 5 minutes, or put it under the broiler for just a minute or 2. Cut into wedges or squares and serve hot, warm, or at room temperature.