Slip one large or two smaller shallow sheet pans into the oven. Preheat the oven to 450°.
Place 5 quarts salted water in a 6-quart pot. Bring to a boil on the stovetop.
In a big bowl, toss together all the ingredients for the roasted vegetables (listed above down to the pasta). Be generous with the salt and pepper.
Pull out the oven rack holding the sheet pan(s). Taking care not to burn yourself, turn the squash blend onto the hot sheet pan(s) and spread it out. Bake for 25 minutes, or until the squash is tender, turning the vegetables two or three times during roasting.
As the squash becomes tender, drop the pasta into the boiling water and cook it until tender, but with some firmness to the bite. Drain in a colander.
Once the squash is tender, turn on the broiler to caramelize it. Watch the vegetables closely, turning the pieces often. Anticipate about 5 minutes under the broiler. You want crusty brown edges on the squash and wilted, almost crisp greens.
Scrape everything into a serving bowl. Add the half-and-half, hot pasta, and 1 cup of the cheese. Toss to blend, tasting for salt and pepper. Add more cheese if desired. Serve hot.