Note: Giardiniera is a pickled vegetable mix that can be found in Italian delis or jarred in supermarkets. If unavailable, it can be omitted. Substitute 2 to 3 seeded and stemmed pepperoncini in its place. This recipe will produce far more olive salad than you need for one sandwich. Extra salad can be stored in the refrigerator for up to a month. Use it in more sandwiches, classic muffulettas, to top pizzas or burgers, or to stir into pasta sauces.
For the Olive Salad: 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 vineger and olive oil and stir to combine. For best results, let olive salad rest overnight before continuing to step 2.
For the Sandwich: Melt one third of butter in a large non-stick skillet over medium heat until foaming subsides. Add both bread slices and cook, swirling occasionally, until pale golden brown on bottom side, about 2 minutes.
Transfer to a cutting board toasted-side-up. Place one cheese slice on top of one slice of bread. Spread a layer of olive salad across bread with a spoon, then top with soppressata, mortadella, and capicola, followed by second cheese slice. Close sandwich, with both toasted sides facing inwards.