Growing up in New York I would often order a Meatball Parmesan hero (grinder, sub, sandwich depending on your part of the world). What I would get was a bunch of meatballs covered with melted parmesan cheese on a long roll. I’ve live for 30 or more years in New Hampshire and here I am likely to be asked “What kind of cheese would you like on that?” It’s as if I hadn’t already said what sort of cheese I was expecting.
Parmesan is almost never an option by the way. American, provolone, mozzarella or even cheddar are more likely options than parmesan. It’s as is Parmesan were a generic word for cheese. But it’s not. No one asks what sort of meat do you want on your Peperoni Pizza do they? Of course not. That would be silly.
Now I don’t mind provolone or mozzarella cheese on my meatball sandwich but why not call it what it is – meatball with cheese?