Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving 1/2 cup of the pasta cooking water.
While the pasta is working, in a small saucepan, heat the stock and saffron. Keep warm.
In a large skillet, heat the EVOO, 2 turns of the pan, over medium-high heat. Crumble in the sausage and cook until browned. Add the pumpkin (or squash), onion, garlic and chile; season with salt and pepper. Partially cover and cook for 7 minutes. Stir in the saffron stock and tomatoes. Cook, stirring occasionally, until reduced, 15 minutes.
Add the pasta and reserved cooking water to the sauce and toss. Serve in bowls and top with the cheese.