This take on the classic couldn't be easier: Instead of dredging and pan-frying, just stack the ingredients in two stages on a baking sheet and broil. Done this way, the tomatoes get lightly roasted and the bread crumbs stay nice and crunchy.
Turn the broiler to high; put the rack 6 inches from the heat. Put 2 tablespoons olive oil on a rimmed baking sheet and spread it around; put the baking sheet in the broiler.
Core and slice the tomatoes.
Slice about 2 pounds large eggplant crosswise 1 inch thick.
Carefully remove the baking sheet from the broiler. Spread out the eggplant slices, but not the tomatoes and turn to coat them in some oil and sprinkle with salt and pepper.
Broil until softened and browned in places, about 3 to 5 minutes. Flip the eggplant, then top with the tomatoes.
Grate the mozzarella and Parmesan.
Strip 16 to 20 basil leaves from the stems.
Combine the bread crumbs, mozzarella, and Parmesan in a small bowl.
When the eggplant is cooked through, remove the baking sheet from the broiler. Lay the basil leaves on top of the tomatoes, sprinkle with the bread crumb and cheese mixture, and drizzle with 3 tablespoons olive oil.
Return to the broiler, and cook until the bread crumbs and cheese are browned and bubbly, 2 to 4 minutes.