The best grilled cheese sandwiches have that stretchy cheese moment when you pull the two halves apart, and that requires a really good melting cheese. You can get this with American cheese, but it is so processed that I recommend using mozzarella instead. Since mozzarella has such a mild flavor, I like to pair it with regular cheddar cheese to add a tanginess to balance out the flavor of the soup. A mixture of cheddar cheese and mozzarella is just right!