Heat the oil in a medium skillet over high heat. Add the corned beef and cook, stirring with a wooden spoon, until it releases some fat and browns slightly, about 3 minutes. Stir in the onion, bell pepper and potatoes and cook, undisturbed, until brown and crisp on the bottom, about 6 minutes. Continue cooking, turning the hash as it browns evenly, about 15 more minutes.
Meanwhile, heat the butter in a nonstick skillet over medium-high heat. Fry the eggs sunny-side up or over easy; season with salt and pepper.
Place the cheese slices on top of the hash, reduce the heat and let sit until the cheese melts, about 1 minute. To serve, top each portion of hash with a fried egg.
Recipe from Crooked Creek Saloon & Eatery, Fraser, CO for Food Network Magazine