Place mushrooms, onion and vinaigrette in a large resealable plastic bag. Seal bag; toss well to coat. Refrigerate 30 minutes. Drain and reserve marinade.
For grill, coat grill rack with cooking spray before starting the grill. Wrap marinated onion slices in heavy-duty foil and place on medium-hot grill. Place mushrooms, gill side down, on grate. Grill both for 5-7 minutes; turn. Grill 5-7 minutes longer or until juices accumulate in the mushroom caps. Remove onions and mushrooms from grill; drain and reserve any juices from onions and mushrooms.
For microwave, place onion slices and mushrooms, gill side up, on a microwave-safe plate. Cover and microwave for 3 minutes. Let stand for 30 seconds. Drain and reserve any juices.
To serve, place each mushroom cap, gill side up, on a serving plate. Layer each with tomato, onion slices and basil. Sprinkle each stack with cheese and thyme. Combine reserved vinaigrette and cooking juices; drizzle over each stack. Season with salt and pepper. Serve warm.