Stand the squash on the large end. Use a large knife or cleaver and cut in half from the stem to the bottom. If you have a rubber mallet, you can use it to tap the blunt side of the blade to help push the knife through the squash. Place the cut sides down on the prepared baking sheet and roast until the skin can be easily pierced with a knife, about 45 minutes.
Remove from oven and set aside until it is cool enough to handle. Use a spoon to scoop out the flesh and put it into a bowl. Mash the squash until it's smooth.
In the 3-quart pot, heat the olive oil and saute the shallots and garlic and cook for 3 minutes, stirring, until they soften. Add the mashed squash, sage, stock, and milk. Reduce heat to medium-low and cook until the sauce thickens, about 10 minutes. Stir frequently. Season with salt and pepper to taste and keep warm while cooking the pasta.
Reduce oven heat to 350° F.
In the 8-quart pot bring water to a boil. Cook the pasta in the boiling water until al dente, not too soft. The pasta will absorb some of the liquid from the sauce in the oven, so don't overcook it. Drain well and put back into the pot.
Pour the sauce over the pasta in the pot, stirring to mix well. Pour sauce and pasta into the prepared baking pan. Sprinkle grated Parmesan cheese on top. Bake for 15 minutes, until the sauce is a little bubbly and the cheese is a light gold color. Serve hot.