In a nonstick pan, heat the oil over medium heat. Add the mushrooms and pepper; cook 3 - 4 minutes until they start to soften. Add in the ham and cook for 1 - 2 minutes more.
Pour the eggs into the pan. As the eggs set around the edge of the pan, with a spatula, gently lift the egg to allow uncooked egg to flow underneath. When eggs are almost set on the surface but still look moist, add the cheese. Slip the spatula under the omelet and fold over.