3 tablespoons extra-virgin olive oil, plus more for drizzling
2 large beefsteak tomatoes, sliced 1/4 inch thick
Salt and freshly ground pepper
1 large onion, thinly sliced
2 garlic cloves, minced
2 teaspoons chopped thyme
One 1 1/2-pound eggplant, sliced lengthwise 1/4 inch thick
Two 1/2-pound zucchini, sliced lengthwise 1/4 inch thick
2 tablespoons chopped parsley
Preheat the oven to 350°. In a large skillet, heat 1 tablespoon of the olive oil. Add half of the tomatoes, season with salt and pepper and cook over high heat for about 10 seconds per side. Transfer the tomatoes to a large plate; repeat with the remaining slices.
Add the remaining 2 tablespoons of oil to the skillet. Add the onion, garlic and thyme, cover and cook over moderate heat, stirring occasionally, until softened. Season with salt and pepper; remove from the heat.
Oil a 9-by-13-inch ceramic or glass baking dish. Arrange one-third of the eggplant on the bottom of the dish, drizzle with oil and season with salt and pepper. Scatter half of the onions on top, followed by half of the zucchini. Drizzle the zucchini with oil, season with salt and pepper and sprinkle with half of the parsley. Arrange half of the tomatoes and their cooking juices over the zucchini. Repeat the layering, ending with a final layer of eggplant. Drizzle with oil; season with salt and pepper.
Cover the casserole with foil and bake in the upper third of the oven for 1 hour. Remove the foil, tilt the baking dish and baste the top of the casserole with the juices. Continue baking for 30 minutes longer, until very tender.
Preheat the broiler. Tilt the baking dish and pour off all but 1/4 cup of the juices. Broil the casserole 4 inches from the heat until browned. Let rest for at least 15 minutes, then serve.