Preheat oven to 375 degrees. Grease a 9x13 baking dish with non-stick cooking spray.
Melt 1 tablespoon butter in the microwave or over the stove. Brush squash with butter. Sprinkle with 1/2 teaspoon salt and 1/4 teaspoon cinnamon. Place on a rimmed baking sheet, flesh-side down and roast until very soft, about 40-45 minutes.
While the squash roasts, make the bechemel and cook the spinach. Add remaining butter to a large saucepan or heavy-bottomed pot. Put the flame on medium-high and cook until the butter begins to brown. Swirl the pan until the butter is a deep brown and smells nutty. Reduce the heat to low and add sage. Cook for 1 minute. Whisk in flour. Cook another minute. Slowly whisk in milk, 1/2 teaspoon salt and nutmeg. Bring to a boil and reduce to a simmer until thickened, about 8-10 minutes. Season to taste with salt and pepper.
Add spinach to a large dry skillet. Turn the heat on medium and cook until wilted, about 3-4 minutes, use tongs to make sure spinach doesn't burn. Season with a little bit of salt and pepper. Toss.
Once the squash finishes cooking, scoop out the flesh and transfer to a bowl. Use a fork to mash up the squash and then season with salt and pepper.
To assemble: Spread 1/4 of the béchamel in the bottom of your prepared baking dish. Cover with three lasagna noodles. Spread with 1/3 of the squash mixture, followed by all of the spinach. Cover with another 1/4 of the béchamel, followed by a 1/3 of the parmesan cheese. Layer with 3 pasta sheets, followed by 1/3 of the squash. Cover with another 1/4 of the béchamel and a 1/3 of the cheese. Cover with three noodles, spread with remaining squash. Cover with remaining béchamel and sprinkle with remaining parmesan cheese.
Cover with foil and place in the oven and bake for 35 minutes. Remove foil and bake another 10-20 minutes or until noodles are soft.