How to Make It
Preheat the oven to 400°. Spread the bread crumbs on a large rimmed baking sheet and drizzle with 1 tablespoon of oil; toss well. Bake for 8 minutes, until the bread crumbs are crisp and golden brown. Transfer the bread crumbs to a medium bowl and toss with the basil and cheese.
Light a grill. Brush the eggplant slices with oil and season with salt and pepper. Grill over moderate heat until charred and tender, about 2 minutes per side. Transfer the eggplant to a platter, fanning them out. Brush the tomatoes with oil and season with salt and pepper. Grill over moderate heat until charred and hot, about 1 minute per side.
Arrange the tomato slices over the eggplant and drizzle with oil. Sprinkle the bread crumbs over the top and serve right away.