Cut the eggplant into 2/3-inch cubes; transfer to a colander, toss with 1/2 tsp. salt and let drain for 20 minutes. Cut the zucchini into bite-size pieces, then cut the onion and bell pepper into 1/2-inch cubes; transfer to a bowl with the garlic.
Position a rack in the center of the oven and preheat to 400 degrees . Wrap the eggplant in a kitchen towel and pat dry. In a nonstick skillet, heat 2 tbsp. EVOO over medium heat. Add the eggplant and cook until halfway done, 8 minutes. Add the zucchini and cook until crisp-tender, 5 minutes. Transfer to a 9-inch square baking dish.
In the same skillet, heat 1 tbsp. eVOO over medium heat. Add the onion, bell pepper and garlic, season with salt and pepper. Cook, stirring occasionally, until crisp-tender, about 5 minutes. Stir in the tomatoes and 1/4 cup parsley and cook until tender, about 5 minutes. Spoon onto the eggplant, then top with the mozzarella.
In a shallow bowl, toss the breadcrumbs, 2 1/2 tbsp. parmesan and 1/4 tsp. each salt and pepper. In a separate shallow bowl, beat the egg. Working with 1 chicken breast at a time, coat the smooth sides with the breadcrumb mixture, then with the egg, then with more breadcrumbs. Place the chicken breasts crumb side up on a large plate.
Clean the skillet and place over medium heat with the remaining 2 tbsp. EVOO. Add the chicken breasts crumb side down and cook until the breadcrumbs are golden-brown, about 5 minutes. Season the uncoated sides, then flip and cook for 1 minute. Transfer the chicken crumb side up to the baking dish and bake until cooked through, about 25 minutes. Let stand for 10 minutes. Sprinkle wiht more parmesan and the remaining 2 tbsp. parsley.