Make the vegetables: Preheat the oven to 425 degrees F. Toss the squash with 1 tablespoon olive oil on a baking sheet; season with salt. Toss the onion with the remaining 1 tablespoon olive oil on a separate baking sheet; season with salt. Roast, rotating the baking sheets once, until the vegetables are tender, 18 to 20 minutes; set aside. Reduce the oven temperature to 350 degrees F.
Meanwhile, make the bechamel: Melt the butter in a medium saucepan over medium heat. Add the sage and cook 1 minute. Whisk in the flour until smooth; cook, whisking, until the flour is lightly golden but not brown, about 2 minutes. Gradually whisk in the milk until smooth. Bring to a boil, then reduce the heat to medium low and simmer, stirring occasionally, until thickened, about 5 minutes. Stir in the nutmeg.
Assemble the lasagna: Butter a 9-by-13-inch baking dish. Toss the cheeses and the flour in a bowl; set aside 1/2 cup for topping. Evenly coat the bottom of the dish with 1/2 cup of the bechamel. Lay 3 noodles side by side on top. Cover with one-third of the remaining bechamel. Sprinkle with half of the remaining cheese mixture, then top with half of the vegetable mixture. Arrange 3 more noodles on top and repeat the layers (bechamel, cheese, vegetables). Top with the remaining 3 noodles, then cover with the remaining bechamel; sprinkle with the reserved 1/2 cup cheese.
Loosely cover the dish with foil and bake until bubbly, about 45 minutes. Turn on the broiler. Uncover the lasagna and broil until browned, about 2 minutes. Let stand 15 minutes before serving.