Combine olive oil, vinegar, basil, bay leaves, garlic and salt in a medium bowl. Stir until well combined.
Place tomato halves cut-sides down directly on greased broiler pan or grill rack. Broil until they start to char and blister (or shrivel and show good grill marks if on the grill), about 4 minutes. Using tongs or a spatula and spoon, carefully turn tomatoes over and broil or grill 2 minutes more.
Remove cooked tomatoes from heat and place cut-sides up in the dish with oil and vinegar mixture. Spoon some of the mixture over tomatoes. Set aside to marinate at least 1 hour, or cover and refrigerate up to 5 days. Serve at room temperature.