Whisk the eggs until frothy and season with salt and pepper.
Meanwhile, melt 1 tablespoon of the butter in a 10-inch nonstick skillet over medium heat. Add the ham and scallions and cook until the ham is browned and warmed through, then remove to a plate.
Wipe the pan out and place it back over the heat. Add the remaining 1 tablespoon of butter and swirl around until melted, then add the eggs. Using a rubber spatula, move the outside edges of the eggs inward as they begin to set up. Continue to cook until the eggs are set and begin to get some color. Sprinkle the ham and cheese evenly over the surface of the eggs. When ready to serve, fold the omelet in half and slide out on to a plate.