In a 12 inch skillet over medium heart, brown beef with onion, green pepper, 1/2 teaspoon of salt and pepper. Remove meat mixture from skillet and drain fat. Reduce heat to low. Melt butter; spread potatoes over butter and sprinkle with remaining salt. Top with beef mixture. Combine eggs and milk; pour over all. Cook, covered, until potatoes are tender, about 30 minutes.
Top with cheese; cover and heat until cheese is melted, about 5 minutes. Cut into wedges or squares to serve.