Spread a thin layer of mayonnaise (or butter) onto one side of each slice of bread. To build each sandwich, gently place the mayonnaise sides of two slices together and down on a clean surface. Carefully layer the shredded cheese and apples together on the bread, making sure to start and end with a layer of cheese.
Melt 1 tablespoon of butter in a skillet over medium heat. Transfer the top slices (the one loaded with cheese and apples) to the pan then top with the remaining slices, mayonnaise facing up. Cook for 1 minute then reduce to medium-low and continue to cook for another 2 minute or until the cheese is beginning to melt.
Flip each sandwich and add the remaining butter to the pan to melt. Cook for another 2-3 minutes or until each side is crisp and golden.