As I understand it, the grinder comes from somewhere between New Haven and New London. It's a cold sandwich on a hero-y roll, made with cold cuts, or tuna salad, provolone, sliced iceberg lettuce and tomato. Oil and vinegar dressing is customary, but not mandatory.
A good grinder doesn't need much protein. The bread has to be very fresh. And the soul of the sandwich is the generous pile of shredded lettuce that cushions the other contents. At a grinder shop near Niantic (perhaps the epicenter of grinder country), in the mid-sixties, the woman behind the counter told me that the sandwich itself was named for "the ground-up lettuce".
That means a hot grinder isn't really a grinder, no matter what they call it.