Place artichoke hearts in a food processor; pulse to chop. Drizzle rolls with 1/2 cup dressing. Divide turkey evenly among rolls; repeat with remaining cold cuts and cheese. Top each hero with 2 Tbsp. chopped artichoke. Divide red peppers, onion and pepper rings among sandwiches. Season with salt and pepper.
Wrap each sandwich in aluminum foil and keep refrigerated or in a cooler until it's time to grill.
Place heroes over indirect heat on grill until they feel warm inside, 10 to 15 minutes. Serve with remaining 1/2 cup dressing.