Step 1

Combine olives, capers, peppers, parsley, giardiniera, and garlic in the bowl of a food processor. Pulse to chop until no pieces larger than 1/2-inch remain. Transfer to a bowl. Add vinegar and olive oil and stir to combine. For best results, let olive salad rest overnight before continuing to step 2.

Step 2

Split muffuletta roll in half and spread each cut surface generously with olive salad, making sure to include the juices when spreading. Layer half of soppressata on bottom half of bun, followed by half of mortadella, half of capicola, and half of provolone. Repeat layers with remaining meat and cheese. Close sandwiches and press down gently to compress. For best flavor, wrap tightly in paper or plastic and let rest for 1 hour before serving. Cut into triangular wedges to serve.