Bring a large kettle or pot of water to a boil. Season the chicken and sausage with 2 teaspoons salt, 1 teaspoon pepper and the marjoram. Heat the olive oil in a heavy-bottomed pot over high heat. Add the chicken and sausage and brown on all sides, about 5 minutes. Transfer to a plate with a slotted spoon.
Add the onions, garlic, bay leaves and tomato puree to the pot. Reduce the heat and cook, stirring, until the oil turns deep red and the onions are tender, 6 to 8 minutes. If the onions are sticking, add a splash of water and scrape the bottom of the pan with a wooden spoon.
Add the spaghetti to the onion mixture and stir-fry until golden, about 6 minutes. Add the chicken and sausage and enough boiling water to cover the pasta by 1/2 inch. Simmer, stirring once or twice, until the pasta is al dente and the sauce thickens, about 10 minutes. Season with salt and pepper and garnish with cheese, if desired.