1. Heat a large grill pan over medium-high heat. Place zucchini in a shallow dish. Add 2 teaspoons oil and garlic; toss to coat. Arrange zucchini in grill pan; cook 2 minutes on each side or until grill marks appear. Cut each zucchini piece in half crosswise. Return zucchini to shallow dish. Drizzle with vinegar. Sprinkle with salt and black pepper.
2. Brush bottom halves of rolls with the remaining 2 teaspoons oil. Top evenly with zucchini, basil, tomatoes, and mozzarella.
3. Brush cut side of roll tops with remaining liquid from shallow dish, and place on sandwiches. Heat the sandwiches in pan until warm.