Combine the oil and flour in a large cast-iron pot or enameled cast-iron Dutch oven, over medium heat. Stirring slowly and constantly for 20 to 25 minutes, make a dark brown roux, the color of chocolate.
Season the onions, bell peppers, and celery with the salt and cayenne. Add this to the roux and stir until soft, about 5 minutes. Add the sausage, the dried thyme and oregano, and cook, stirring often, for 5 minutes. Add the broth and bring to a boil. Reduce the heat to medium-low and simmer, uncovered, for 45 minutes. Add the reserved turkey meat and the reserved onions and celery and cook for 15 minutes. Add the parsley and green onions.