Split rolls horizontally and lightly brush cut sides of bottom halves with a little oil. Layer them with shredded cheeses, prosciutto, and onion. Spread 1/4 cup relish on each top half and set on sandwiches.
Heat a heavy medium frying pan, preferably cast iron, over medium heat. Lightly brush pan with oil. Add sandwiches and reduce heat to medium-low. Set a pie pan on sandwiches and top with 4 medium cans of food. Cook, gently pressing down on cans every so often, until sandwiches are golden brown on underside, 4 to 7 minutes.
Remove pie pan and cans. Brush sandwich tops with a little more oil and sprinkle with parmesan. Turn over, replace pie pan and cans, and cook, pressing on cans again, until cheese is melted and second side is nicely browned, 2 to 3 minutes.
Cut sandwiches in half and tuck in arugula. Serve with remaining relish.