Preheat one side of the grill to medium high heat and brush the avocado halves with 1 tablespoon of oil. Grill until you get nice dark grill marks.
To a bowl, mix together the tomatoes, 3 tbsp of basil, the remaining oil and salt and pepper, to taste.
Stuff the avocado shells with the tomatoes and add a few pieces of the sliced mozzarella cheese. Place on the side of the grill without direct heat (that is off) and close the lid. Melt the cheese for 1-2 minutes.
Transfer to a platter, and top with the parmesan cheese, balsamic glaze, and additional basil.