Heat a large heavy skillet or griddle over medium heat. In a large bowl, stir together ham, potatoes, green pepper, and onion. Melt 2 tablespoons of the butter in the large skillet or generously butter the large griddle; spread skillet evenly with hash mixture or mound onto griddle and flatten slightly.
Cook over medium heat for 8 to 10 minutes or until brown; turn occasionally.
Meanwhile, melt remaining butter in a large skillet over medium heat or use the same griddle. Add eggs; cook until whites are just set. Turn and cook for 1 to 2 minutes more for soft eggs or 3 to 4 minutes more for firm. Yolks should be just set.
Spoon 1/4 of the hash onto each warm plate; top with an egg, flipping it "over easy." Makes 4 servings.
If you like, use refrigerated shredded hash brown potatoes or frozen shredded hash brown potatoes, thawed, in place of the chopped cooked potatoes.