Preheat grill to medium-high. Brush eggplants, cut into 1/2-inch thick rounds, with 4 tablespoons olive oil; sprinkle with salt and pepper. Grill eggplant until grill marks form (2-3 minutes); flip, and grill until tender (3-4 minutes more). Remove and drizzle with balsamic vinegar. Cut baguette into 4 pieces; slice each in half. Brush bottoms of baguette evenly with 1 tablespoon plus 1 teaspoon olive oil. Slice mozzarella into 8 slices; layer half of the slices among sandwiches. Top each sandwich with 1/2 cup fresh basil leaves, 3-4 slices eggplant, remaining cheese, and baguette tops. Press sandwiches in a panini press or in a large skillet over medium-high heat for 3 minutes per side.