Preheat oven to 400 F. Toss squash, oil and 1 teaspoon salt on a baking sheet. season with pepper. Bake until light gold and tender, 25 to 30 minutes. Let cool.
Reduce oven temperature to 375F. Combine ricotta, yolks, cream, mozzarella and a pinch of nutmeg in a medium bowl. Season with salt.
In a small bowl, mix together the pumpkin, cinnamon nutmeg, maple syrup. Season with salt and pepper (to your taste); set aside.
Melt butter in a small saute pan over medium-high heat. As soon as it starts to sizzle, add sage and cook until light gold and slightly crisp at edges, 3 to 4 minutes.
Place squash in a medium bowl and mash 1/2 of it (I used a potato smasher, or use a fork) and leave 1/2 in full pieces. Fold in the sage butter and 3/4 cup of the broth. If you feel it needs more broth add a tablespoon more at a time. You don’t want it to be too soupy. Season with salt and pepper.
Spoon 1/3 of the pumpkin sauce in a 9 cup baking dish. Top with a layer
of noodles. Spoon 1/4 of the ricotta mixture over the noodles. Top with a layer of noodles. Top with 1/4 of the butternut squash mixture. Top with 1/3 cup of the pumpkin sauce.
Top with a layer of noodles. Top with a layer of the ricotta cheese. Top with a layer of noodles, squash and pumpkin sauce.
Spread the last bit of ricotta mixture on top, dotting with mozzarella pieces and grated Parmegiano Reggiano.
Cover with foil. Place baking dish on a rimmed baking sheet and bake for about 45 minutes, until the cheese is bubbling and golden. Let stand 15 minutes before serving.