Prepare a barbecue (or preheat a broiler). Cut the eggplants into slices about 1/2-inch thick. Spread on a baking sheet and brush with olive oil on both sides. Set aside.

Cut mozzarella in half crosswise, then slice each half about 1/4-inch thick to create half-moon shaped pieces. Slice tomatoes in the same manner. Arrange tomatoes and mozzarella decoratively around the edge of a large serving platter.

Place eggplant slices on barbecue (or place on baking sheet in broiler) and cook, turning once, until nicely charred on each side; the eggplant should almost caramelize. Arrange the cooked eggplant down the center of the serving platter, surrounded by the tomatoes and mozzarella. Pluck about 12 large leaves from the basil bunch, stack them, roll them up like a cigar, then slice crosswise into thin shreds (?chiffonade?). Scatter the basil shreds over the vegetables and cheese, then drizzle balsamic vinegar and more olive oil over all. Sprinkle with salt and pepper, to taste, and let stand until eggplant is at room temperature.