Prepare barbecue (medium-high heat) or preheat broiler. Cut eggplants on diagonal into 1/2-inch-thick slices. Brush slices lightly with olive oil. Grill or broil eggplant until golden brown on both sides, turning once, about 4 minutes. Cool eggplant slightly.
Whisk 1/4 cup olive oil, vinegar and garlic to blend in small bowl. Layer eggplant in baking dish, seasoning eggplant with salt and pepper, drizzling with oil mixture and sprinkling with chopped basil between each layer. Let eggplant stand 1 to 2 hours at room temperature before serving.