Which cheeses melt
Age, moisture, and acid decide whether a cheese stretches, pools, or holds its shape.
Good melters
Young pressed cheeses and stretched-curd cheeses melt. Gruyère, young Gouda, fontina, Monterey Jack, and low-moisture mozzarella are reliable. They have enough moisture to flow and not so much age that the proteins seize.
Cheeses that do not melt
Paneer, halloumi, queso fresco, and very fresh acid-set cheeses keep their shape in a pan. A long-aged grating cheese such as Parmigiano-Reggiano softens but does not stretch. Use it as a finish, or mix a little into a sauce.
A smoother melt
Heat the cheese gently. A hard boil makes the fat split from the protein. For a sauce, grate the cheese and add it off the boil. Processed American cheese melts in a sheet because emulsifying salts keep the fat and protein together. That is a different product from a wheel of cheddar.