1. Preheat grill to medium-high. Combine cheese, green onions, shallot, salt, and basil in a medium mixing bowl. Season with freshly ground pepper.
2. Using a small sharp knife, remove a cone-shaped piece 2 inches wide and 1 inch deep from the top of each tomato. Fill cavities with cheese mixture, dividing equally.
3. Place tomatoes directly on grill rack or on aluminum foil placed on grill rack. Cover barbecue with lid. Cook until tomatoes are soft, about 5 minutes.