Preheat the oven to 250 degrees ;. Divide the mozzarella among 4 pork cutlets; top with the remaining 4 cutlets. Season with salt and pepper. Place the flour, eggs and breadcrumbs in three separate shallow bowls. Coat the pork stacks in the flour, then in the eggs, then in the breadcrumbs.
In a large skillet, heat 1/4 cup olive oil over medium-high heat. Add 2 pork stacks and cook, turning once, until golden, 4 to 6 minutes. Transfer to a paper-towel-lined baking sheet and keep warm in the oven; repeat with 3 tablespoons olive oil and the remaining 2 pork stacks.
Wipe out the skillet and return to the heat. Add the remaining 1 tablespoon olive oil, the garlic, broccoli and 2 tablespoons water; season with salt. Cook until crisp-tender, 3 to 5 minutes. Serve with the pork stacks.