Bring a large pot of well-salted water to a boil over high heat. Put the rigatoni in the boiling water and cook until just shy of al dente, about 10 minutes.

While the pasta cooks, heat 1/2 tablespoon of the oil in a 12-inch skillet over medium-high heat. Add the sausage and cook, breaking it into pieces with a spatula or spoon, until it's almost cooked through, 3 to 5 minutes. Using a slotted spoon, transfer the sausage to a bowl. Pour the fat out of the skillet but do not wipe it clean. Heat the remaining 2-1/2 tablespoons oil in the skillet over medium heat and cook the shallots until they begin to soften, about 1 minute. Raise the heat to medium high and add the squash. Cook, stirring frequently, until the squash is barely tender, 3 to 5 minutes.

Reserve 1/2 cup of the pasta-cooking water and drain the rigatoni. Return the rigatoni to its cooking pot and add the sausage, the squash mixture, and 2 tablespoons of the reserved pasta water. Toss over medium heat until the sausage is cooked through and the rigatoni is perfectly al dente, about 3 minutes. Add more of the pasta water as necessary to keep the dish moist.

Remove from the heat, add the goat cheese and parsley, and toss until the cheese melts and coats the pasta. Season to taste with salt and pepper, transfer to warm shallow bowls, and top each serving with some of the grated Parmigiano, if using.