Combine first 3 ingredients in a small bowl; stir well, and set aside. Combine tomato and next 3 ingredients in a medium bowl; stir well, and set aside.
Remove brown gills from the undersides of mushrooms with a spoon; discard gills. Coat top and bottom of mushroom caps evenly with cooking spray; sprinkle with salt and pepper.
Coat grill rack with cooking spray; place rack on grill over mediumhot coals (350° to 400°). Place mushrooms, top side up, on rack; grill, covered, 4 minutes. Turn mushrooms over; spoon tomato mixture evenly into caps; top each mushroom with 1 tablespoon basil mixture and 2 tablespoons cheese. Grill, covered, 4 minutes or until mushrooms are tender and cheese melts.