Beat the eggs in a bowl until well combined. Mix in 2/3 of the cheese, milk, ham, salt and pepper. Heat the butter in a large, non-stick skillet over medium heat. Pour in the eggs. As mixture begins to set, gently move spatula across the bottom and sides of the skillet to form large, soft curds. Cook until the eggs are thick but still moist, with no visible liquid egg remaining. Divide English muffins among 4 plates. Top with scrambled eggs. Top the eggs with remaining cheese. Garnish with green onions or rosemary.