Heat the 1 tablespoon of the oil in a small nonstick skillet. Add the ham, bell pepper, and onion and sauté 5 minutes or until the ham is browned and the vegetables are tender. Set aside.
Whisk the eggs, salt, and pepper in a medium bowl. Heat half of the remaining oil in the skillet over medium heat. Add half of the egg mixture and let cook, undisturbed, for about 30 seconds or until set. Using a rubber spatula, lift up the sides of the egg in several places, allowing uncooked egg to flow underneath.
Once the eggs are set, sprinkle the top with half of the cheese. Top one half with half of the ham and vegetable mixture. Carefully fold the cheese only side of the omelet in half over the ham mixture. Slide the omelet onto a serving plate. Repeat with the remaining eggs, cheese, and ham mixture.