Teleport yourself to a sidewalk cafe in Paris with these chicken paillards in a spicy-sweet tomato sauce.
Pound chicken between pieces of plastic to a scant 1/4 inch thick. Heat grill to medium-high. Combine 1/4 cup oil and garlic in a large cast-iron skillet; place on grill. Cook until golden, about 1 minute. Add tomatoes and Peppadews; season with salt and pepper. Cook until mixture thickens slightly and tomatoes burst, about 12 minutes. Remove skillet from grill; cover grill to reheat, 5 minutes.
Drizzle chicken with 1 tablespoon oil; season with salt and pepper. Grill until charred in spots, 1 to 2 minutes. Flip, top with cheese, and cover grill. Cook until cheese melts and chicken is cooked through, 1 minute. Transfer to a plate; spoon tomato mixture over top. Top with basil. Dress arugula with remaining 1 tablespoon oil and Peppadew brine. Serve atop chicken, with bread.