In a medium bowl beat eggs, milk, salt, and pepper until blended. Heat 1 teaspoon of butter in a large non-stick skillet over medium heat. Pour in the egg mixture. As the eggs begin to set, gently pull them across the pan with a spatula, forming large curds. Continue cooking while pulling and folding the eggs until thickened no and the visible liquid egg remains. Remove from the pan, set aside. Clean out the skillet.
Heat the skillet to medium heat. Spread the remaining 2 teaspoons of butter evenly on one side of each slice of bread. Place 2 slices in the skillet butter side down. Top each with 1 slice of cheese, scrambled eggs, bacon, and an additional slice of cheese. Cover with the remaining bread slices, butter side up.
Cook sandwiches over medium heat until browned on the bottom. Once they are browned on the bottom flip them over and cook until browned on the other side and the cheese has melted.