Preheat oven to 400°F.
Peel squash, cut in half lengthwise, scoop out and discard seeds, slice into ½-inch cubes, and toss in a large bowl with 2 tablespoons olive oil. Spread out cubes in a single layer on a sheet tray, season with salt and pepper, and roast in the preheated oven for about 30 minutes until tender and golden.
Bring a large pot of water to a boil and cook lasagna noodles for about 5 minutes, stirring so they don’t stick together. Remove noodles to a bowl of cold water until ready to assemble lasagna.
For the ricotta filling: In a medium bowl, combine ricotta, lemon zest, salt, pepper, and nutmeg.
For the olive oil béchamel sauce: Heat 2 tablespoons extra virgin olive oil in a medium saucepan over medium heat. Stir in flour and cook, stirring, for about 3 minutes until smooth and bubbling. Whisk in the milk and simmer, whisking until the mixture begins to thicken. Turn down the heat and continue whisking until thickened, about 10 minutes. Season with salt and pepper and set aside.
Sauté garlic in 1–2 tablespoons olive oil for about 1 minute or until it just starts to color. Stir in spinach and cook just until the spinach wilts. Remove from heat and set aside.
Coat the bottom of an 8-inch square baking dish with olive oil and lay down a layer of noodles. Add a layer of roasted squash cubes and wilted spinach, then a layer of ricotta and some grated parmesan. Cover with noodles. Add another layer of squash, spinach, ricotta, and parmesan. Top with another layer of noodles and cover with béchamel sauce. Arrange the fresh sage leaves over the sauce and sprinkle on any remaining parmesan. Bake for 30 minutes. Remove from oven and allow to cool 10 minutes. Slice and serve drizzled with olio nuovo or new-harvest olive oil.