Melt 1 tablespoon butter in frying pan; saute onion and green pepper, stirring often, until tender.
Combine eggs, milk, salt and pepper.
Pour over onion and green pepper in frying pan and cook over low heat, stirring often to let uncooked portion of eggs flow beneath cooked portion.
Continue cooking until eggs are set but still moist.
Divide eggs into six portions.
Spread toasted muffin halves with remaining tablespoon of butter, and top muffin halves with egg mixture and a slice of cheese.
Broil until cheese melts and is lightly browned (about 5 minutes).