Mix biscuit baking mix, milk, salt, pepper, and eggs in a bowl. Stir in chopped ham, chopped green bell pepper, and sliced green onions.
Heat butter in a skillet until melted and bubbly. Gradually pour the egg/ham mixture into the skillet. Cover and cook over medium-low heat until almost set, about 10 to 12 minutes.
Sprinkle with shredded cheddar cheese; cover and cook until the cheese is melted, about 3 minutes longer.