Preheat a grill or broiler to medium-high. Brush each eggplant round on both sides with EVOO and season with salt and pepper. Grill until tender, about 5 minutes (or broil for 10 minutes).
Grill the 4 bread pieces until lightly charred on both sides. Rub with the garlic, brush with EVOO and season with salt.
Layer each piece of bread with the grilled eggplant, tomato slices, basil leaves and mozzarella slices. Grill the sandwiches, covered, until the cheese is melted, 1 to 2 minutes.