Cook the spaghetti in a large pot of well-salted boiling water according to the package directions for al dente. Drain.
While the pasta cooks, heat the oil in a large skillet or Dutch oven over medium heat. Add the onion; cook and stir until tender, about 5-7 minutes. Add the garlic and stir for 1 more minute.
Stir in the pumpkin, chicken broth, cream, Parmesan cheese, parsley, rosemary, nutmeg, salt, and pepper. Heat through, whisking frequently.
Add the cooked, drained spaghetti and the chicken to the sauce. Cook over medium heat, just until warmed through. Taste and season with additional salt and pepper, if desired.
Serve on individual plates and garnish with fresh herbs or extra Parmesan cheese.