Cheese lovers question me about great melting cheeses. As a rule, not all cheeses are appropriate for melting: cottage cheese, paneer, feta, ricotta, and halloumi. These cheeses don't melt because their curds are milk protein. Protein doesn't melt or spread, so don't consider these cheeses for sauces.
Which cheeses melt?
Manchego, mild cheddar, American, taleggio, Comte, gruyere, fontina, Monterey jack, provolone, raclette, parmesan, bries (remove the rind), blues, and mozzarella are the best melting cheeses.
Aged dry cheeses, with little to no moisture, should be added in your Mornay recipe before other cheeses since they require additional time to break down on low heat. For example, if you make a cheese sauce using, parmesan reggiano, cream cheese, and gruyere. You would add the parmesan first into your bechamel, and let it melt for ~10-15 minutes because it needs time to break down from having no moisture to being reconstituted to be broken down. Than you would add your gruyere and let it melt for ~5-7 minutes since it has a bit of moisture, then closer to the end before you take the sauce off the heat, you would add the cream cheese, since it will take 2-3 minutes to melt.
Chef, do you prefer whole pieces or shredded cheeses?
I recommend whole pieces of cheese to shredded cheeses since prepackaged bits don't melt and become stringy. Cheesemakers coat shredded cheeses to prevent caking, which prevents them from melting. Pre-shredded cheeses can turn your cheese sauce into a wiry mess; instead, get one block of cheese, shred it by hand or appliance, and enjoy!
Great tip: Cheese is fresh if it melts like low-moisture mozzarella with a long cheese strand. If the strand breaks at a short distance, like an inch or two, the cheese is old or getting old. When cheese gets old, it's not bad, but it will lose its flavor. But low heat is what is considered to give balance since many melting cheeses have different degrees and times in melting, but if its low heat and you have time to monitor you can manage by aged, semi-hard, and fresh cheeses. Just by altering the times, it can help give your sauces some depth.