In a medium bowl, whisk together eggs, ⅛ teaspoon salt, and ⅛ teaspoon pepper. Set aside.
Add ½ tablespoon butter to a nonstick 7- or 8-inch nonstick skillet over medium heat.
Add ham, green pepper, onion, ⅛ teaspoon salt, and ⅛ teaspoon pepper to the skillet and cook until veggies are softened and ham is just starting to brown, about 3-5 minutes. Remove to a bowl and set aside.
With skillet still over medium heat, add remaining ½ tablespoon butter and let coat bottom of pan.
Pour the beaten eggs into the skillet and let them cook undisturbed for a couple minutes until they start to set around the edges.
Using a spatula, gently push the cooked edges toward the center, tilting the pan to let the uncooked eggs flow to the edges to cook. Continue until there’s no liquid egg left, but the top is still slightly runny. You can gently flip the omelet at this point, but you don't have to.
Add cooked veggies and ham to one side of the omelet and sprinkle with shredded cheese.
Carefully fold the omelet in half, covering the filling and let cook for another minute to melt cheese.