Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the spaghetti, and return to a boil. Cook the pasta uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, about 12 minutes.
Drain well in a colander set in the sink.
Heat the olive oil in a skillet over medium heat. Stir in the garlic and onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Stir in the chicken stock, pumpkin puree, and half-and-half. Season with hot pepper sauce, cinnamon, nutmeg, salt, and pepper. Reduce heat to medium-low and simmer until sauce has thickened, about 4 minutes. Toss goat cheese into pumpkin sauce and heat through.