In a large, heavy pot place the hen, water, quartered onions, celery pieces, bay leaves, 1 tablespoon of the salt, and 1 teaspoon of the cayenne pepper and bring to a boil over high heat. Reduce the heat to medium and cook, partially covered, until the hen is tender, about 2 hours. Remove the hen and set aside. Strain and reserve the broth.
In a large, heavy pot or a Dutch oven over medium heat, combine the oil and flour. Cook, stirring constantly, until the roux is a dark, chocolate brown color, about 20 to 25 minutes. Add the chopped onions, bell peppers, celery and chopped sausage. Cook, stirring, until the vegetables are very soft, about 8 to 10 minutes. Add the reserved chicken broth and stir until the roux mixture and broth are well combined. Bring to a boil, then reduce the heat to medium-low and cook, uncovered, stirring occasionally, for 1 1/2 hours.
Meanwhile, remove the skin from the hen and pick the meat off the bones, discarding the skin and bones. Coarsely chop or tear the meat into bite size pieces. Add the chicken and the sliced sausage to the gumbo. Cook for 15 minutes. Remove from the heat and let sit for 5 minutes before skimming any fat that has risen to the surface.
Stir in the green onions and parsley and serve the gumbo in individual soup or gumbo bowls over hot, steamed white rice.
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