1. Prepare a grill for medium heat (350-450). Combine 1T oil and the garlic and brush onto 1 side of each bread slice.
2. Lay bread oiled side down on grill and cook until slightly toasted, about 2 minutes. Turn bread over, lay tomato slices on bread to fit, overlapping if needed, then lay cheese slices over tomatoes. Cover grill and cook until cheese starts to melt, about 4 minutes.
3. Transfer sandwiches to a platter. Put 2 basil leaves over each sandwich and drizzle with remaining T oil and the vinegar. Sprinkle with salt and pepper.