Here's the situation: You're headed to the beach—or the park, or a music festival, or a bonfire (?!)—with a group. You need something you can make in advance that will withstand the long journey and impress the friends who only thought to bring Swedish Fish and a bag of popcorn. Here's the solution: This sandwich, which looks like it came from the Italian deli of your dreams and which you'll definitely want to eat whilst in a beach chair, watching the waves hit the sand (or the dogs scamper in the park etc.). While we love the trifecta of mortadella, soppressata, and cooked ham, feel free to get creative with the meats you choose. Just remember to get a mix of cured and cooked for an array of flavors and textures.
Prep your veg! Cut 1 head of iceberg lettuce in half through the core. Peel off flimsy outer layers. Reserve one half for another use (perhaps this Iceberg Salad with Italian Dressing?). Cut remaining half in half again through the core (so you have two quarters), then thinly slice crosswise to create shreds. Peel ½ small red onion and cut in half through the root. Thinly slice each wedge lengthwise so that you make wispy arcs. Cut 1½ cups pickled peppadew peppers in half.
Drain and finely chop 1 cup jarred mixed giardiniera. Aim for the veg to be the size of peas—you don't want huge pieces that will get in the way of your sandwich experience. Transfer to a small bowl. Stir in ⅔ cup mayonnaise; taste and season with salt.
Slightly overlap 2 sheets of parchment paper to create 1 larger sheet. Cut 16x4" loaf sesame bread in half lengthwise with a serrated knife and place on parchment. You will build the sandwich on parchment for easy wrapping and cutting.
Spread giardiniera mayonnaise on cut sides of bread with a large spoon.
Layer ⅓ lb. thinly sliced cooked ham, ⅓ lb. thinly sliced mortadella, ⅓ lb. thinly sliced soppressata, and ⅓ lb. sliced provolone on bottom half of bread, folding or draping as necessary to keep it tidy and within confines of bread.
Spread onion and peppadew peppers over meats and cheese.
Top with shredded lettuce. It will seem like a lot and you may not think it's possible to pack it all on—but have a little faith and don't hold back. Once you close up the sandwich, the lettuce will compact down quite a bit and you'll be grateful you piled on more than you thought was necessary.
Drizzle ¼ cup pickled pepper brine over lettuce. (Acidic, spicy, and slightly sweet, pickle brine is one of our favorite stand-ins for vinegar or lemon juice.) Season lettuce generously with salt and pepper. Gather up overhang of parchment paper and securely wrap. Cut in half with your serrated knife. Make a sawing motion (rather than a pressing one) so that you're not squishing your sandwich.
Wrap entire hoagie tightly in plastic wrap, then cut crosswise into 4 even sections. (The plastic wrap acts sort of like a girdle, holding in all of the meat, cheese, and veg so that they won't fly out when you slice smaller segments. It also acts as an insurance against leaking.) If you're taking your sandwich out into the world, wrap again in plastic or slip back into bread bag for easy transporting.