In small bowl, combine vinegar, garlic, 1 tablespoon mint, oregano, salt and pepper. Whisk in oil until thick and well blended.
Cut eggplant into twelve 1/2-inch-thick slices. Arrange in 13 x 9 x 2-inch glass baking dish; coat with dressing.
Heat grill. Grill pitas, cut side down, until toasted, 1-1/2 minutes. Remove to tray. Grill eggplant until tender, about 3 minutes per side.
Place eggplant slice on each pita half. Drizzle any remaining dressing over eggplant. Sprinkle with cheese.
Grill pitas over indirect heat, with grill covered, until cheese is just warmed and slightly melted, about 2 minutes. Remove to platter. Sprinkle with additional mint, if desired. Serve immediately. Makes 6 servings.
