Preparing the squash:
Peel the using a sharp knife. (You can alternatively use a vegetable peeler, but I find that a knife is quicker.) Pitch seeds, or set aside to roast at a later date. Cut the squash into 1/2 inch cubes, toss with olive oil, and season with salt and pepper. Add squash to pan, and cook over medium-high heat until squash starts to caramelize at the edges, about 5 minutes.

Once the squash starts to caramelize, add 3/4 cup water and simmer over medium-low heat for 5 minutes.
If you have the time (or have the ability to plan ahead, a quality I do not possess) you could roast the squash instead of on the stove. Roasting the squash would really enhance the flavor of the dish, and would be pretty hands-off. Cube the squash as mentioned above, season with salt and pepper and toss with olive oil, and roast at 400 degrees until cooked through.

Transfer cooked squash to a food processor and pulse until smooth. If you have a small food processor, you may want to do this in two batches.

Making the Bechamel:
Melt butter over medium heat in a large saucepan. Once the butter has melted, add the flour and whisk for approximately 1 minute, or until the butter and flour are incorporated and the raw flour smell is gone. Gradually whisk in the milk, and bring to a boil.and simmer for 5 minutes, or until the sauce coats the back of a spoon. Whisk in the nutmeg. Transfer half of the sauce to a blender, and add the basil. Blend basil and bechamel until smooth. Return the basil sauce to the rest of the bechamel in the saucepan and stir to incorporate. Season sauce with salt and pepper to taste.

Note: When blending hot liquids, always make sure that you crack the lid of the blender while blending, otherwise your hot liquid can build up and burst out of the blender container.

Assembly:
Preheat oven to 375 degrees. Put a layer of bechamel in the bottom of the pan, and cover with 1 layer of noodles. Spread squash puree over the noodles, and sprinkle with cheese. Repeat sauce, noodles, squash, 3 more times. (Note: You can stop here, cover with foil, and freeze for up to a month if you prefer.)

Tightly cover lasagna with foil and bake for 40 minutes. After 40 minutes, sprinkle on the parmigianno-reggiano and remove the foil; allow to cook for an additional 15 minutes uncovered, until the top of the lasagna is golden brown. Let stand for 15 minutes before serving. Serves eight as a main course.