In a deep 12-in. cast-iron or other heavy skillet, melt butter. Add the potatoes, sausage, onion and green pepper. Cook, uncovered, over medium heat until potatoes are lightly browned, 15-20 minutes, turning once., Push potato mixture to the sides of pan. Pour eggs into center of pan. Cook and stir over medium heat until eggs are completely set. Season with salt and pepper. Reduce heat; stir eggs into potato mixture. Top with cheese; cover and cook until cheese is melted, 1-2 minutes.