In a large, non-stick skillet about 10 inches in size, heat half of the oil over medium/low heat.
Add half the veggies and ham, sprinkling with some of the salt and pepper. Cook for about 3 minutes or until starting to brown.
Meanwhile, mix 3 eggs and the remaining salt and pepper in a small bowl.
Transfer the veggies and ham to a clean plate. Do not wipe out the pan.
Pour the mixed eggs into the pan. Rotate the pan to ensure an even layer of eggs.
Allow the eggs to cook for a minute without touching.
Use the edge of a spatula to loosen the edges of the eggs and rotate the pan again to allow the uncooked egg to touch the pan. Cook for an additional minute.
The eggs should be just about set and no longer glossy, add ¼ of the shredded cheese. Cover with a lid and continue to cook for an additional minute.
Remove the lid and set aside. Sprinkle the veggies and ham over half of the omelet.
Use the spatula to lift the edge of the omelet to fold it over the veggies.
Transfer the omelet to a clean plate by sliding it out of the pan. Sprinkle with another ¼ of the cheese and serve warm.
Repeat with the other half of the ingredients to make the second omelet