Preheat oven to 375ºF. Add tomatoes and the 4 crushed garlic cloves to a baking dish (we used 9x13 inch), toss with 2 tablespoons olive oil, balsamic vinegar, sugar (if using) and season with salt and pepper. Roast, turning once or twice, until tomatoes are soft, and skins blister and split, about 30 - 40 minutes. Set aside and reserve with any tomato liquid that may have collected. (Tomatoes can be roasted a day ahead)

Prepare a gas or charcoal grill for high heat cooking (a grill pan on the stovetop will also work).

Chop off the top and bottom ends and peel the eggplants partially in long vertical stripes. Slice eggplants into 1/2-inch thick rounds. Brush each round generously with olive oil (about 3 to 4 tablespoons, total) on both sides and season with salt and pepper. When grill is ready, cook until eggplant is soft and golden brown on both sides, about 5 minutes per side. If eggplant seems dry, brush on a little more oil while cooking. Remove from grill to a plate, cover loosely with foil and set aside. (Eggplant can be grilled a day ahead)

On the stovetop, add 3 tablespoons oil and the minced garlic to a cold 8-inch skillet. Turn heat to medium and cook until the garlic sizzles and turns fragrant but not brown, about 1 minute. Add breadcrumbs and toss in the garlicky oil. Cook, stirring often, until crumbs turn golden brown, 4 minutes. Transfer crumbs to a medium bowl, let cool for a few minutes and then stir in herbs and parmesan cheese. Season with salt and pepper to taste.

Assemble dish by adding a layer of eggplant to a platter. Top with a layer of tomatoes, and a drizzle of the tomato cooking juices. Then add another layer of eggplant and tomatoes until all are used. Drizzle again with collected tomato juices. Nestle in chunks of burrata cheese, and sprinkle all over with garlic and herb breadcrumbs. Add a few basil leaves, if desired. Serve warm with warm, crusty bread and good, rich red wine.