Melt butter in a medium non-stick saucepan over medium-high heat.
Add the onions, peppers, and ham to the skillet and cook until the onions and peppers are tender.
Pour them into a bowl and set aside.
Whisk together the eggs, half and half, salt, and pepper.
Melt the remaining 2 tablespoons of butter in the same skillet over medium-low heat.
Pour the egg mixture into the skillet.
Gently move the eggs a little bit at a time in the pan so that the cooked eggs move to the top and the uncooked eggs can reach the bottom of the skillet.
As the eggs cook more and more, stir less and less so that they form a solid layer.
Once the eggs are almost all the way cooked through, sprinkle the cheese and the cooked ham and veggie mixture in an even layer over the top of them.
Cover with a lid until the eggs are cooked through (160ºF) and the cheese is melted.
Uncover and fold the omlete in half.
Slide onto a plate and cut into 4 sections for serving.