In a small bowl, stir together the mayonnaise and butter. Spread the mixture on one side of each slice of bread.
Heat a large cast-iron skillet or griddle over low heat. Add 4 pieces of the bread, buttered side down. Top each piece of bread with ⅔ cup of the shredded cheese. Top each sandwich with a second piece of bread, mayonnaise side facing up.
Cook, watching closely, until golden brown on the bottom, about 8 minutes. Flip and cook on the other side until the cheese is melted and both sides are golden brown, about 6 more minutes.
Transfer the sandwiches to plates, cut in half and serve warm.
