Remove the innards (liver, heart, and gizzard) of the chicken. In a small saute pan, heat 1 tablespoon olive oil and add the innards to the pan and saute over high heat until they are browned. Season with salt and pepper and chop coarsely.
Bring 2 quarts water to a boil.
In a Dutch oven, heat 3 tablespoons olive oil over high heat and add the chicken. Brown it all over, then remove from the pan, and set aside.
In a large bowl, combine the reserved innards, bread crumbs, cheese, egg, garlic, parsley, nutmeg, and salt and pepper, to taste. Mix well and stuff the cavity of the bird with the stuffing. Truss the bird with butcher's twine and place in the Dutch oven, along with the celery and onions.
Pour 2 quarts boiling, salted water over the chicken, and place over medium flame. Cook meat for 1 hour or until the chicken is falling apart and the cooking liquid has mingled with the chicken's juices. Serve the broth hot with chicken pieces.