I got this recipe from Bon Apetit a few years ago, and it is a summertime favorite. If you don't have garlic oil on hand, just add a half teaspoon of garlic powder to the cheese mixture and use plain olive oil instead.
Stir cheese mix, breadcrumbs, and 1/3 cup basil in medium bowl to blend.
Prepare barbecue (medium heat). Sprinkle eggplant and tomatoes with salt and pepper. Brush 1 side of eggplant slices with some oil. Grill, oiled side down, until bottom side is tender and grill marks appear, about 3 minutes. Brush second side of eggplant slices with more oil; turn over and top grilled side with tomatoes, then cheese mixture. Sprinkle with Parmesan. Drizzle with remaining oil. Cover and grill until tomatoes are tender and cheese melts, about 7 minutes. Transfer eggplant slices to plates. Sprinkle with remaining 2 tablespoons basil and serve.