Crack whole egg gently into a small ramekin. Place egg white in a second ramekin. Beat with a fork for 30 seconds and set aside.
Melt 2 tablespoons butter in a nonstick pan over medium heat. Add ham, onion, and bell pepper. Cook until ham is heated through and vegetables are tender, 2 to 3 minutes. Transfer to a bowl.
Melt remaining butter in the nonstick pan over medium heat. When butter bubbles, pour egg into the center of the pan. Pour egg white in a circular motion around the outside of the egg to increase overall size. Quickly sprinkle ham mixture and Cheddar cheese over egg whites before they set. Cover pan and cook until egg is cooked to desired doneness and cheese is melted, 2 to 5 minutes. Serve immediately.
