Heat coals or gas grill for direct heat. In large shallow dish, toss asparagus, mushrooms and dressing. Place vegetables in grill basket (grill “wok”).
Cover and grill vegetables over medium heat 8 to 12 minutes, shaking basket or stirring vegetables occasionally, until asparagus is crisp-tender. Sprinkle vegetables with basil and cheese. Cover and grill 3 to 4 minutes longer or until cheese is slightly melted.