Meanwhile, in small (1-quart) saucepan, heat butter on medium for 2 to 3 minutes, or until browned and fragrant, swirling often. Stir in flour. Cook 1 minute, stirring continuously. Gradually whisk in milk until smooth. Add remaining 1/4 teaspoon salt and nutmeg. Continue cooking, stirring constantly, until the sauce comes to a simmer and is thickened. Remove from heat.
In a medium skillet, heat remaining 1 tablespoon oil over medium-high heat. Add the onion and cook, stirring, for 3 to 4 minutes, until the onion begins to brown. Reduce heat to low and continue to cook, stirring frequently, until the onion is deeply golden brown, about 8 to 10 more minutes. Add the balsamic vinegar and 1 tablespoon water and cook, scraping the bottom of the pan to lift any browned bits, until liquid is evaporated. Remove from heat.
Working in batches, cook pasta in a large pot of boiling salted water, stirring occasionally, until just softened, about 30 seconds. (If using dried noodles, cook until al dente.) Transfer noodles to a large rimmed baking sheet as you go. Do not overlap.
Grease a 9- by 4-inch loaf pan or small baking dish. Place one lasagna sheet in the bottom of the dish and top with 1/3 of the squash and 1/3 of the onions. Spread 1/4 of béchamel sauce over squash, then top with 1/3 of Fontina. Repeat layering twice. Top with the remaining lasagna sheet. Spread remaining sauce over the pasta and sprinkle with Parmesan.
Bake 25 to 30 minutes on the middle rack of the oven, until top is golden brown. Sprinkle hazelnuts over the top. Let stand 10 minutes before serving.