1/3 cup long-grain white rice
1 1/2 lb summer squash, sliced crosswise 1/4 inch thick
6 1/2 tablespoons olive oil, divided
1/2 lb small tomatoes, sliced crosswise 1/4 inch thick
1 medium onion, halved lengthwise and thinly sliced
3 garlic cloves, finely chopped
2 large eggs, lightly beaten
1 teaspoon chopped thyme
1/2 cup grated Parmigiano-Reggiano, divided
Preheat oven to 450°F with racks in upper and lower thirds.
Cook rice according to package instructions. While rice cooks, toss squash with 1 tablespoon oil and 1/2 teaspoon salt in a shallow baking pan. Toss tomatoes with 1/2 tablespoon oil and 1/4 teaspoon salt in another baking pan. Roast squash in upper third of oven and tomatoes in lower third, turning vegetables once halfway through roasting, until tender and light golden, about 10 minutes for tomatoes; 20 minutes for squash. Leave oven on. Meanwhile, cook onion and garlic with 1/2 teaspoon salt in 2 tablespoons oil in a large heavy skillet, covered, over low heat, stirring occasionally, until very tender, 15 to 20 minutes. Stir together onion mixture, cooked rice, eggs, thyme, 1/4 cup cheese, 1 tablespoon oil, ¼ teaspoon salt, and 1/2 teaspoon pepper. Spread half of rice mixture in a shallow 2-quart baking dish, then top with half of squash. Spread remaining rice mixture over squash, then top with remaining squash. Top with tomatoes and drizzle with remaining 2 tablespoons oil, then sprinkle with remaining 1/4 cup cheese. Bake in upper third of oven until set and golden brown, about 20 minutes.