1. For the cheese: In a medium shallow bowl, combine 1/4 cup olive oil with the basil, garlic, salt, and pepper. Add the mozzarella, and toss to coat well. Cover bowl, and marinate 2 hours.

2. Slice each eggplant into eight wedges by cutting in half lengthwise and cutting each half into quarters; transfer to a medium bowl. In a separate bowl, whisk together vinegar, water, sugar, and remaining 1/2 cup olive oil. Pour over eggplant, and turn to coat; cover, and transfer to refrigerator for 1 hour.

3. Preheat broiler. Remove eggplant from marinade, and shake off excess liquid. Place on baking sheet, and broil until crisp and browned, about 10 minutes. Transfer to paper towels to drain.

4. With tomatoes still on the vine, place on one end of a rimmed baking sheet. Remove mozzarella from marinade; place on baking sheet beside tomatoes. Drizzle tomatoes with marinade, and sprinkle mozzarella and tomatoes with Parmesan cheese. Broil until tomatoes burst and mozzarella has melted, about 6 minutes.

5. To serve, transfer tomatoes and mozzarella to serving platter, and place eggplant spears on the side. Garnish with balsamic vinegar.