Spread butter or mayo on one side of each of the 4 bread pieces. Heat a griddle pan to medium high heat. Place 2 pieces of bread on the griddle, buttered side down. On each piece, spread 1/4 cup mozzarella cheese, a sprinkle of shredded Parmesan, tomato slices, avocado slices, the other 1/4 cup mozzarella cheese, Parmesan, a sprinkle of kosher salt, and the other piece of bread (spread with mayo if you’d like).
When the bottom bread is browned, flip the sandwich and cook until the bread is toasted and the cheese is melted.