Preheat broiler. Rub the eggplants and garlic with olive oil and prick eggplants all over with a knife or fork. Place eggplants and garlic on a rimmed baking sheet and broil until garlic is browned and tender (remove if it is done before the eggplant) and eggplants are charred and shriveled, turning a few times, about 30 minutes. Let the eggplants cool about 15 minutes. Meanwhile, place peppers skin-side up on a baking sheet and coat lightly with olive oil. Broil about 5 to 9 minutes, until the skins just start to brown. Remove. Turn oven to 375 degrees and adjust rack to middle position.

Peel the garlic and eggplant skins and discard, placing flesh in a large bowl. If liquid accumulates in the bowl, drain most of it out before proceeding. Using an immersion blender (or mashing by hand), make a thick puree. Squeeze ¼ lemon over the mixture and stir in basil, panko or bread crumbs, salt, black pepper to taste, and 1 heaping tablespoon of the Parmesan cheese (save the rest for the topping). Taste and adjust seasonings as needed.

Coat a baking dish with olive oil. Arrange the peppers hollow side up in the dish. Spoon in the eggplant mixture. Top with a slice of Roma tomato slightly overlapping a piece of burrata or mozzarella. Spread the remaining Parmesan over the top and sprinkle lightly with black pepper.

Bake for 15 to 20 minutes, until the cheese is melted and the tomatoes tender. Serve warm, garnished with more fresh basil if desired.