Preheat oven to 225ºF and line a heavy cookie sheet with aluminum foil.
In a bowl, toss cherry tomatoes with brown sugar, olive oil, salt, and pepper.
Arrange tomatoes cut side up on the cookie sheet with their juices and roast for 45 minutes, toss, then roast another 30 to 45 minutes until slightly shriveled and sticky.
While tomatoes roast, combine breadcrumbs, Za'atar, and Parmesan cheese in a bowl; lightly whisk eggs in another bowl.
Sprinkle eggplant slices generously with salt on both sides, layer them on paper towels, and let sit for 30 minutes to draw out moisture.
Rinse salted eggplant slices thoroughly and pat dry.
Heat 1 to 2 tablespoons vegetable oil in a large skillet over medium heat until shimmering.
Dip each eggplant slice first in beaten egg, then coat in breadcrumb mixture.
Fry eggplant slices 3 minutes per side until golden brown, then drain on paper towels.
To assemble, spread a small amount of whipped ricotta on a plate as a base, then layer eggplant, ricotta, roasted tomatoes, and their juices in three layers.
Drizzle remaining tomato juices over the layered stacks and garnish with fresh thyme leaves.