Place stuffed olives, Kalamata olives, red onions, lemon juice, artichoke hearts, basil, olive oil, capers, and garlic in a food processor. Pulse until coarsely chopped. Refrigerate olive salad 8 hours to overnight.
Spray a panini press with cooking spray and preheat the grill.
Layer 1 slice provolone cheese, 1 spoonful olive salad, 2 slices ham, 2 slices salami, another spoonful olive salad, and 1 additional slice provolone cheese on 4 slices bread; top with 4 additional slices to create 4 sandwiches. Place sandwiches in the preheated panini press and grill until bread is toasted, 3 to 5 minutes each. Cut sandwiches in half.