Preheat oven to 500°F and adjust racks for upper third of the oven. Line 3 baking sheets with foil and set aside.

Place eggplant slices in a single layer in a large colander or on top of a paper towel-lined surface. Sprinkle kosher salt liberally over eggplant and set aside for 30 minutes. Meanwhile, roast the peppers.

To Prepare Roasted Peppers:

Using a pastry brush, brush inside pepper halves with about 1-2 teaspoons olive oil. Sprinkle with kosher salt, and arrange in a single layer skin-side up on bak-ing sheet. Roast on rack closest to heating element for about 15 minutes or until skin appears charred. Remove from oven and cover with foil; set aside to cool. When cool enough to handle, separate charred skins from peppers and discard. Dice the skinned peppers and set aside. 

To Prepare Roasted Eggplant:

Blot salt-soaked eggplants with paper towels to absorb excess moisture. Then, using a pastry brush, brush slices lightly with olive oil (about 2 tablespoons oil, or more as needed). Transfer eggplant slices to 2 prepared baking sheets, arranging them in a single layer. Roast eggplant in upper third of the oven until tender, rotating trays halfway through cooking – about 8-10 minutes. Remove from oven; set aside to cool. Reduce oven to 300°F.

To Prepare Tomato Base:

Heat about 3 tablespoons olive oil in a large, deep skillet over medium-high heat. Add onion and garlic; season with about 1 teaspoon kosher salt and stir to blend. Sauté until onions are translucent, about 5-6 minutes. 

Break up the canned tomatoes with your hands or a knife and add with juices to the pan. Add bay leaves, 1 teaspoon thyme and season with freshly ground black pepper. Continue cooking for another 6-7 minutes — mixture should be slightly thickened. Add reserved diced peppers and balsamic vinegar. Stir to blend; cook for another 1-2 minutes. Remove from heat. 

To Assemble:

Divide tomato base between the bottoms of 2 oven-to-table casserole dishes; spread in an even layer. Layer 1 slice of roasted eggplant, zucchini, yellow squash, and plum tomato in an overlapping pattern over tomato base. Continue pattern, forming rows or concentric circles until casseroles are full. 

 2. Season to taste with salt, pepper, and remaining 1 teaspoon of thyme. Drizzle with remaining 3 tablespoons olive oil. Cover with foil or parchment and bake for 30 minutes. Raise temperature to 350°F, uncover, and bake for another 30 minutes. 

 3. Remove from oven and sprinkle basil and parsley over top. 

Can be made up to 2 days in advance.