For the roasted tomatoes: Preheat the oven to 375°.  Place the tomatoes and the crushed garlic cloves in a roasting dish.  Toss with about 2 tablespoons of olive oil, balsamic vinegar, sugar and salt and pepper.  Roast, gently stirring a few times throughout the cooking process, until the tomatoes are soft and begin to split, about 30 to 40 minutes.  Remove from the oven and set aside.

For the eggplant: Preheat a grill pan or outdoor grill over high heat.  Cut off the top and bottom of the eggplant and slice the eggplant into 1/2-inch-thick rounds.  Brush both sides of the eggplant with olive oil and season with salt and pepper.  When the grill has preheated, cook the eggplant until tender and golden but not falling apart, about 3 to 5 minutes per side.  Remove from the grill to a plate and cover loosely with foil.

In the meantime, heat a small skillet over medium heat; add 3 tablespoons olive oil and the minced garlic and cook until fragrant, about 1 minute.  Add the fresh breadcrumbs and cook, stirring often, until the crumbs are golden brown and crisp, about 4 minutes.  Transfer the breadcrumbs to a small mixing bowl and stir in the basil, parsley and Parmesan cheese.  Season with salt and pepper to taste.

Layer the eggplant on a large serving platter.  Top with the tomatoes; pull the burrata into pieces and arrange on top.  Sprinkle the garlic-herb breadcrumbs all over the top.  Serve warm or at room temperature.