Cut chicken into serving pieces.
Put the neck, giblets, and backbone in a pan, pour in enough water to cover, and add 1 onion (cut in half), 1 stalk of celery (cut in chunks), a few sprigs of parsley, 1 teaspoon salt, 1/4 teaspoon pepper, and 1/2 teaspoon dried thyme.
Cook over a low heat about 45 minutes.
Strain and reserve the broth.
Combine flour, salt, and pepper in a paper bag.
Drop in pieces of chicken and shake until evenly coated.
Heat butter in a skillet (one with a tight fitting lid), add chicken, and saute until nicely browned on all sides.
Add broth, cook to a boil, cover tightly, and simmer gently for 1 to 1 1/4 hours or until chicken is tender when tested with a fork.
Transfer chicken to an 8 inch square baking dish that can be used for serving.
Reserve the broth.
To prepare the batter topping: Sift together flour and salt.
Set aside.
Beat eggs soundly, pour in milk and butter.
Stir in flour mixture until smooth.
Pour over the chicken, covering it evenly with the batter.
Bake in a preheated 450° oven for 15 minutes.
Reduce oven temperature to 350° and continue baking 20 to 25 minutes.
Topping will puff up and brown around the edges like Yorkshire Pudding.
While the pudding bakes, make a smooth paste with 1 or 2 tablespoons of flour and water.
Stir into chicken broth and cook, stirring constantly, until slightly thickened.