Heat barbecue to medium heat, approximately 325 degrees F. While that is preheating, clean your mushrooms and remove the stems.
In a medium bowl, mix together the tomatoes, garlic and balsamic vinegar.
Spray a vegetable grilling tray well with oil. Gently spoon the tomato mixture into the mushroom caps (you're going to want to discard as much liquid as possible) and arrange on the tray.
Grill for 8-10 minutes, until mushrooms are tender and cooked through.
Immediately place a small spoonful of goat cheese on the warm mushrooms, scatter the basil leaves, and sprinkle generously with salt & pepper.