Finely chop green olives, black olives and giardiniera and pepperoncini; place mixture in a medium mixing bowl.
Add remaining ingredients (garlic, parsley, basil, salt, pepper, red wine vinegar and olive oil); stir until well combined. Cover and refrigerate at least 2 hours, or overnight.
For the Grilled Cheese
Brush one side of each slice of bread with butter.
Lay two slices buttered-side down and top each with: ¼ cup of the Olive Salad, ¼ of the shredded Monterey Jack cheese, ½ of the soppressata, ½ of the salami, ½ of the Mortadella, another ¼ of the shredded Monterey Jack cheese and another slice of buttered bread (buttered side out).
Preheat a cast-iron skillet or frying pan over medium-low heat.
Add the sandwiches and cook on one side for 2-3 minutes. Flip and continue cooking 2-3 more minutes, or until cheese has melted and sandwich is golden brown. Cut each sandwich in half and serve immediately.
Notes
You will have lots of leftover olive salad. Extra salad can be stored in the refrigerator for up to a month – use it to make more sandwiches! (It’s also great as a dip with pita chips.) There are several types of giardiniera mixes. “Chicago-style” is quite spicy and features finely chopped peppers and vegetables. It’s delicious, but for this muffaletta olive salad, I recommend using a milder “Italian-style” giardiniera.