Pre-heat a grill or broiler to medium-high. Brush the mushroom caps with EVOO, season with salt and pepper and grill, turning once, until tender, 7-10 minutes.
Meanwhile, using a food processor, pulse the giardiniera and olives into a relish.
Layer the mushroom caps with the sopressata, provolone, hot ham, relish and mozzarella. Grill, covered, or broil the muffuletta pizzas until the cheese is melted, 1-2 minutes.
Dress the greens with the lemon juice and a drizzle of EVOO; season to taste with salt and pepper. Serve with the muffuletta pizzas.