Slice the rolls lengthwise, but not all the way through, creating a hinge. Spread butter or olive oil on the outside crust of each roll. In a small bowl, mix mayonnaise and Italian dressing, then spread this mixture on the inside of both halves of each roll.

On the bottom half of each roll, layer the provolone cheese first (this helps it melt onto the bread). Then evenly layer the Genoa salami, capicola, and mortadella. Top with tomato slices, red onion, giardiniera, and lettuce. Season the tomatoes lightly with salt and pepper.

Close the sandwiches. Heat a large skillet, panini press, or griddle over medium heat. Place the sandwiches in the skillet. If using a regular skillet, place a heavy pan or foil-wrapped brick on top to press them down. Cook for 4-5 minutes per side, until the bread is golden brown and crispy and the cheese is fully melted.

Remove the sandwiches from the heat. Let them rest for 1 minute, then slice each sandwich in half diagonally. Serve immediately while warm and crispy.