Grilled Eggplant Parmesan

1 1/3 cups shredded four-cheese mix or pizza cheese
2 tablespoons Italian-style dry breadcrumbs
1/3 cup plus 2 tablespoons thinly sliced fresh basil
1 medium eggplant (about 1 pound), cut lengthwise into 1/2-inch-thick slices
8 ounces plum tomatoes, thinly sliced
1/4 cup garlic-flavored olive oil
2 tablespoons freshly grated Parmesan cheese

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.