Melt the butter in a skillet over medium heat. Cook the onion, garlic scapes, and ham in the melted butter until the onion is tender, about 5 minutes.
Beat the eggs, milk, salt, and pepper together in a small bowl; pour into the skillet and cook, stirring continually, until the eggs have nearly set, but are still slightly moist. Add about half the Cheddar cheese; cook and stir until the eggs are completely cooked, about 5 minutes. Top with the remaining Cheddar cheese to serve.