Bring a large pot of salted water to a boil. Cook the lasagna noodles 3 to 4 minutes less than the package directs. Drain, reserving 1 cup pasta water, and pour the pasta onto a baking sheet to cool. Drizzle with olive oil to prevent the pasta from sticking and set aside until ready to assemble.
For the Bolognese: Place a heavy-bottomed pan over medium-high heat and add the oil. Add the beef, season with salt, and cook, breaking up any lumps, until it begins to brown. Move the beef to one side of the pan, add the squash and cook until it begins to brown and soften, 2 to 3 minutes. Add the onions, celery and garlic with a large pinch of salt and cook, stirring occasionally, until the vegetables soften and begin to caramelize, about 5 minutes. Stir in the tomato paste and toast for a few minutes until it looks rusty. Add in the white wine, scraping the brown bits from the bottom of the pan with a wooden spoon. Add the crushed tomatoes and thyme bundle and bring to a simmer. Stir in the reserved 1 cup pasta water and cook over low heat, partially covered, until the sauce has thickened and reduced, at least 20 minutes. Season with salt and pepper and remove the thyme bundle.
For the bechamel: Melt the butter in a heavy-bottomed saucepot over medium heat. Whisk in the flour until it forms a wet sandy consistency. Slowly whisk in the milk and bring to a gentle boil. Reduce to a simmer, then whisk in the nutmeg and 1 teaspoon each salt and freshly ground black pepper. Cook, stirring often, until thickened and smooth, 10 to 15 minutes. Remove from the heat and whisk in the ricotta, pecorino and egg yolks, then set aside to cool slightly.
Preheat the oven to 375 degrees F. Line a baking sheet with foil. Place a 7-quart roasting pan or a lasagna pan on the baking sheet.
To assemble: Spoon a layer of Bolognese into the pan, covering the whole bottom. Add an even layer of noodles and press into the sauce. Add a layer of Bolognese over the noodles, then another layer of pasta. Spoon some of the bechamel over the noodles, covering the entire surface. Add another layer of pasta, pressing down gently. Spoon a layer of Bolognese to cover the surface. Repeat the layering, ending with a layer of Bolognese on top. Sprinkle with the mozzarella, then cover with foil and bake until completely warmed through, 45 minutes to 1 hour.
Uncover and continue to bake until golden brown and crusty on top, about 10 minutes. Sprinkle with grated Parmesan and let set 20 to 30 minutes before serving.