Preheat oven to 400° or heat grill to medium. If baking in the oven, grease wide, shallow baking dish or gratin dish with olive oil. Using the tips of your fingers, scoop seed pockets out of tomatoes. Don’t be too obsessive about getting out all juice and seeds. (Discard what you do remove)
In a skillet, heat 2 tablespoons of olive oil with garlic over medium heat until aromatic. Add breadcrumbs and stir 2 or 3 minutes to toast the crumbs. Remove from heat and add salt, pepper and cheese. Use a spoon to fill cavities in tomatoes, then pile more on top until all is used. Place on baking sheet (or on a rack on the grill) and drizzle with 2 tablespoons olive oil. Cook 20 to 25 minutes, until softened and charring. Arrange on a platter and serve immediately. Leftovers are great reheated.