Remove thick end of mushroom stem, leaving about 1/2 inch (1 cm) attached to cap. Place, smooth side up, in large, shallow glass dish. Combine oil, vinegar, thyme, salt and pepper; pour over mushrooms, turning and brushing to coat evenly. Let stand for 15 minutes. Reserving marinade, place mushrooms, smooth side down, on greased grill over medium-high heat; close lid and cook for 5 minutes. Turn and baste with marinade; cook, covered, for about 3 minutes or until mushrooms yield to the touch when pressed. Arrange Parmesan over top; cook, covered, for about 2 minutes or until melted.