Combine chicken giblets, neck, and backbone with onion, celery, parsley, 1 teaspoon salt, thyme, and 1/4 teaspoon pepper in a medium saucepan; add water to cover. Bring to a boil; cover and simmer 45 minutes. Strain and reserve broth. Discard meat and vegetables.
Combine 1/2 cup flour, 1 teaspoon salt, and remaining 1/4 teaspoon pepper. Dredge chicken pieces in flour mixture, coating well; brown in 1/3 cup butter in a large skillet over medium heat. Add reserved broth to chicken. Bring to a boil; cover and simmer 45 minutes or until chicken is tender. Remove chicken from skillet, and cool; reserve broth. Remove meat from bone; dice, and place in a 1 1/2-quart casserole.
Combine eggs, milk, and 2 tablespoons butter; beat well. Add 1 1/4 cups flour and 1/2 teaspoon salt; beat well. Pour batter over chicken; bake at 450° for 15 minutes. Reduce temperature to 350°; bake 20 minutes or until lightly browned.
Stir 3/4 cup water into reserved chicken broth. Combine 1 tablespoon flour and remaining water, mixing well; stir into chicken broth. Cook mixture over medium-high heat, stirring constantly, until thickened and bubbly. Serve gravy with pudding.
Oxmoor House Homestyle Recipes