For the tomato sauce: Process tomatoes, basil, oil, garlic, sugar, salt and red pepper flakes in food processor until smooth, scraping down sides of bowl as needed, about 30 seconds. Transfer sauce to a bowl and set aside. (Sauce can be refrigerated for up to 1 day.)

For the filling: Bring 3 quarts water to boil in a large saucepan. Add cauliflower florets, cashews, and 2 teaspoons salt and cook until cauliflower is very soft and falls apart easily when poked with fork, about 20 minutes. Drain cauliflower mixture in colander and let cool slightly, about 5 minutes.

Process cauliflower mixture, 3 Tablespoons oil, and 1/4 cup water in clean, dry food processor until smooth, scraping down sides of bowl as needed, about 2 minutes (mixture will be slightly grainy). Season with salt and pepper to taste. Transfer 1/4 cup mixture to bowl and stir in remaining 1 Tablespoon oil and basil; set aside for topping. (Mixtures can be refrigerated for up to 3 days.)

For the vegetables: Adjust oven rack to upper-middle position and heat oven to 450 degrees. Toss eggplant and mushrooms with 2 Tablespoons oil, garlic and 1/2 teaspoon salt in bowl, then spread on rimmed baking sheet. Toss zucchini with remaining 1 Tablespoon oil, and 1/4 teaspoon salt in now-empty bowl. Roast eggplant-mushroom mixture until beginning to wilt, about 15 minutes. Remove sheet from oven, stir zucchini into vegetables, and continue to roast, stirring occasionally, until mushrooms are lightly browned, eggplant and zucchini are tender, and most of the juices have evaporated, 15 to 20 minutes. Set aside. (Cooked vegetables can be refrigerated for up to 1 day.)

For the lasagne: Adjust oven rack to middle position and heat oven to 375 degrees. Grease 13 by 9-inch baking dish. Spread 11/3 cups tomato sauce over bottom of dish. Arrange 4 noodles on top. Spread half the cauliflower filling over noodles, followed by half of vegetables. Spread 11/3 cups tomato sauce over vegetables. Repeat layering with 4 noodles, remaining cauliflower filling, and remaining vegetables. Arrange remaining 4 noodles on top, and cover completely with remaining tomato sauce.

Cover dish with aluminum foil and bake until edges are bubbling, 45 to 50 minutes, rotating dish halfway through baking. Dollop lasagne evenly with 8 to 10 spoonfuls of reserves cauliflower topping, and let cool for 25 minutes. Drizzle with oil, sprinkle with remaining 1 Tablespoon basil, and serve.