Put an oven rack in the upper third of the oven. Preheat the oven to 425 degrees F.
Spray a baking sheet, liberally, with vegetable oil cooking spray. Set aside.
Mix the squash and onion together and arrange in a single layer on the prepared baking sheet. Drizzle with olive oil and season with salt and pepper, to taste. Bake for to 40 to 45 minutes until the vegetables are golden and cooked through. Remove from the oven and set aside to cool slightly.
While the squash mixture is cooling, bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and reserve about 2 cups of the pasta water. Put the pasta, goat cheese and 1 cup of pasta water in a large serving bowl. Toss until the cheese has melted and forms a creamy sauce. Add the squash and onion mixture, the walnuts and the basil. Toss well and season with salt and pepper, to taste. Garnish with Parmesan and serve.
To toast the walnuts, arrange them in a single layer on a baking sheet. Bake in a preheated 350 degree F oven until lightly toasted, about 6 to 8 minutes Cool completely before using.