Whisk together the eggs, salt, pepper, and dried parsley until frothy. Set aside.
2 eggs, 1/4 teaspoon salt, 1/4 teaspoon pepper, 1/4 teaspoon dried parsley
Heat a small skillet over medium. Add about 1/2 tablespoon butter. Once the butter is melted, add the ham. Cook, stirring often, until warmed through and starting to brown around the edges, about 3 to 4 minutes. Keep warm.
1 1/2 tablespoons butter, 2 ounces ham
Add the remaining butter to a 8-inch non-stick skillet over medium heat. Swirl to ensure that the melted butter coats the entire bottom of the pan. 1 1/2 tablespoons butter
Pour the egg mixture into the skillet and ensure it covers the bottom of the pan. As the eggs start to set, if there is quite a bit of uncooked egg on top, you can lift the edge of the omelet and tilt the pan to allow it to run off the top and around the edges. Cover with a lid and cook until the eggs are just about set, about 2 minutes.
Reserve about 1 tablespoon of ham and cheese, then place the remaining cheese onto one half of the egg and top with ham.
2 ounces cheddar cheese, 2 ounces ham
Fold the empty half of the egg over the filling. Heat another minute or so to melt the cheese.
Transfer the omelet to a plate and top with the reserved ham and cheese. Garnish with sliced scallions or minced fresh parsley, if desired.