Place the eggplant and zucchini slices on wire racks and sprinkle lightly on both sides with kosher salt. Let stand for 30 minutes, until they have released some of their liquid, then blot them dry with paper towels

Heat a large nonstick skillet over medium-high heat.  Working in batches, brush the slices of eggplant and zucchini lightly on both sides with olive oil and lightly brown in the skillet on both sides. Transfer to a plate as vegetables are browned and repeat with the remaining vegetables. To the same skillet, add enough oil to lightly coat the bottom of the pan and add the sliced peppers. Cook until they begin to soften, and then add the onions and season generously with salt and pepper. Cook, stirring frequently, until vegetables are softened. Add the garlic and cook until fragrant, 2 to 3 minutes. Add the tomatoes and cook until they have released their liquid and it has mostly evaporated. Season to taste with salt and pepper.

Preheat the oven to 350°F and lightly oil a 2 1/2 quart casserole or gratin dish. Spread one-third of the tomato mixture over the bottom of the dish, then top with 1/2 of the eggplant and zucchini slices. Sprinkle with 1 1/2 tablespoons of the basil. Add half of the remaining tomato mixture and then top with the remaining eggplant and zucchini. Sprinkle another 1 1/2 tablespoons of basil over the veggies, then spread the remaining tomato mixture over the top. Sprinkle the remaining tablespoon of basil over the top and drizzle the top lightly with olive oil. Cover the casserole with aluminum foil and bake in the center of the oven for 30 minutes, or until the vegetables are hot throughout and quite soft. Remove the foil and continue baking until the vetables are lightly golden on top. Remove from the oven and set aside to cool briefly before serving. Sprinkle with the parsley and serve either hot, warm, or at room temperature.