For sauce, in a small saucepan melt 1 tablespoon butter. Stir in flour and dash pepper. Add the milk all at once. Cook and stir over medium heat until thickened and bubbly. Cook and stir for 1 minute more. Stir in American cheese until melted. Remove from heat. Set aside.
In a medium skillet melt 1 tablespoon butter. Pour egg mixture into the skillet. Cook over medium heat, without stirring, until mixture begins to set on the bottom and around edge. Using a large spatula, lift and fold the partially cooked egg mixture so that the uncooked portion flows underneath. Continue cooking for 2 to 3 minutes more or until egg mixture is cooked through but is still glossy and moist.
Spoon the eggs into two small casseroles. Sprinkle with the ham and mushrooms. Pour sauce over ham and mushrooms.
Bake, covered, in a 350 degree F oven for 15 to 20 minutes or until heated through. Sprinkle with green onion. Makes 2 servings.