1. Preheat oven to 450 degrees. On a rimmed baking sheet, toss onions, squash, and sage with oil. Season with salt and pepper. Arrange in a single layer and roast until tender, about 25 minutes, rotating sheet and tossing vegetables halfway through.

2. Meanwhile, in a large pot of boiling salted water, cook pasta according to package instructions. Reserve 1/2 cup pasta water; drain pasta and return to pot. Add vegetables and cheese and toss to combine, adding enough pasta water to create a thin sauce that coats pasta.