Cut the squash in half and roast it in the oven for an hour or more on 350F, or until it's soft enough for a fork to go through without resistance. Keep it in the oven with the heat off after it is cooked to keep warm until using.
Boil the pasta water and cook when it's ready.
Make the Alfredo and put it to the side.
Place the shiitake mushrooms in a large oiled skillet and start browning them on medium heat, adding a few tablespoons of water to it every here and there. It will get absorbed quickly and help the mushrooms keep their juiciness without overcooking.
Strain the pasta at al dente, leaving it firm to bite. Leave the pasta water to the side. Empty the into the same skillet as the mushrooms, add the alfredo, and toss it.
Take the squash out of the oven and fill it with the pasta. Garnish with a few of the almonds and chopped fresh basil.