Slice the tomatoes in half. Make sure to slice them across the wider art of the tomato, not through the stem. Sprinkle the cut side of the tomatoes with salt. Place them on a wire rack with the cut side down. Place a bowl or glass baking dish underneath to collect the juices. Let them drain for 15-30 minutes. Save the juices.
Meanwhile, preheat the grill over medium-high heat. Brush the tomatoes with half of the of olive oil. Grill them with the cut side down for 4-5 minutes until they begin to char slightly. Flip and cook for 4-5 minutes until they begin to bubble.
While the tomatoes cook, whisk together the remaining olive oil with tomato juice (from draining the tomatoes), balsamic vinegar, and garlic. Serve this drizzled over the cooked tomatoes. Add fresh herbs like basil, thyme, or parsley, Consider adding some feta, goat, or Parmesan cheese. Season with salt and pepper as needed.