Combine olive oil, garlic, and salt in a small bowl; set aside.

Preheat an outdoor grill to 400 degrees F (200 degrees C) and lightly oil the grate.

Put eggplant slices on the hot grill and cook for 3 to 4 minutes. Turn eggplant and brush with the oil mixture. Place 1 slice of tomato and 1 slice of mozzarella cheese on each eggplant piece. Close the lid and cook until cheese has melted, 3 to 4 minutes. Transfer to a plate and sprinkle with basil.