Toast the English muffin halves until golden brown in a toaster or under a broiler, then set aside on a plate.
Heat a frying pan over medium heat; add butter and let it melt, coating the surface evenly. Place the ham slice in the pan and cook for 1-2 minutes until lightly caramelized and heated through, then transfer to a plate.
1 piece English muffin
Lower the heat to medium-low. Crack the egg into the same pan, season with salt and pepper, and cook until the whites are set but the yolk remains runny, about 2-3 minutes. Use a spatula to gently lift and check for doneness.
1 piece English muffin
Place a slice of cheddar cheese over the hot egg during the last 30 seconds of cooking so it begins to melt, creating a cohesive layer.
1 piece English muffin
Assemble the sandwich by placing the egg and melted cheese on one toasted muffin half, then top with the caramelized ham, and cover with the other muffin half.
1 piece English muffin
Optional: gently press the sandwich to compact everything, then serve immediately while warm and the cheese is gooey and the eggs are tender.