Prepare the olive spread: In a food processor, pulse the giardiniera pickles, roasted bell pepper, black olives, green olives, garlic, lemon zest, and olive oil until they are roughly minced but still chunky.
Slice the focaccia bread horizontally to create a top and bottom layer.
Spread the olive mixture evenly on both sides of the focaccia.
Layer the bottom half of the focaccia with half of the provolone, half of the mozzarella, and all of the soppressata, mortadella, and capicola. Top with the remaining provolone and mozzarella.
Close the sandwich with the top half of the focaccia.
Heat a cast iron pan over medium-high heat.
Brush the sandwich on both sides with more olive oil.
Grill the sandwich pieces on the heated cast iron pan until the cheese melts and the bread is crispy, about 5 minutes per side.
Press from above with another pan to flatten the sandwich.