Make 4 sandwiches by placing 1 layer of the cheese between 2 slices of the bread. Gently mash the sandwiches to seal in the cheese. Heat a large cast-iron skillet or griddle over medium heat. In a shallow bowl, whisk together the eggs and milk and season with the salt and pepper. Melt 1 tablespoon of the butter in the skillet. Dip 2 of the sandwiches in the egg mixture to moisten both sides and place in skillet. Cook until browned and crisp, about 4 minutes per side, pressing with a spatula to seal. Wipe pan clean, add the remaining butter, and repeat with the other 2 sandwiches. Serve immediately.