In a small bowl combine boiling water and dried tomatoes. Let stand for 20 minutes. Drain, discarding water; chop tomatoes.
Trim stem and blossom ends of eggplant; cut eggplant crosswise into 1/2-inch-thick slices. Place slices on a baking sheet; brush both sides of slices generously with the 3 tablespoons oil and sprinkle evenly with the 1 teaspoon salt and the 1/2 teaspoon pepper.
For a charcoal or gas grill, place eggplant slices on the rack of a covered grill directly over medium heat. Grill for 10 to 12 minutes or until slightly charred and tender, turning once halfway through grilling. Transfer grilled slices to a clean plate and let cool to room temperature.
In a food processor or blender combine dried tomatoes, the snipped basil, the 1/4 cup oil, and the vinegar. Cover and process or blend until finely chopped.
On a large platter arrange eggplant slices, mozzarella cheese slices, and tomato slices, overlapping slices. Sprinkle lightly with coarse salt and pepper. Spoon dried tomato mixture on top. Garnish with basil leaves. Serve at room temperature.
