Some butternut squashes are sweeter than others, and there's no way to predict this by appearance. If the squash sauce seems a little bland, add about a teaspoon of sugar. It will brighten the flavor considerably.
1. Cut squash into chunks, and place in food processor. Pulse machine on and off until squash looks grated. (Alternatively, grate or chop the squash by hand.) Set a large pot of salted water to boil for the pasta.
2. Place a large skillet over medium heat, and add the butter or oil. A minute later, add the squash, salt, pepper and about 1/2 cup of water. Cook over medium heat, stirring occasionally. Add water, about 1/4 cup at a time, as the mixture dries out, being careful not to make it soupy. When the squash begins to disintegrate, after about 10 or 15 minutes, begin cooking the pasta. While it cooks, season the squash with the nutmeg, sugar if necessary, and additional salt and pepper if needed.
3. When the pasta is tender, scoop out about 1/2 cup of the cooking liquid, then drain. Toss it in the skillet with the squash, adding the reserved cooking water if the mixture seems dry. Taste, and adjust the salt, pepper or nutmeg as you like; then, toss with the cheese and serve.