1) For dumplings,in med. bowl whisk together egg, cold water, oil and the 1/2 tsp. salt. Stir in flour. Mix until well blended and elastic. Cover refrigerated 2 hrs.
2) Season chicken, including back and neck with 1 tsp. salt. Set aside. In 6- qt. dutch oven combine chicken stock,water,celery, onion,and pinch of kosher salt. Bring to a boil. Add chicken pieces placing leg quarters and backbone first. Place breast,skin side on top. Reduce heat to just below simmer. Cover,leaving half-inch opening.
3)Cook 30 to 45 mins. or until breast are just done; remove. continue cooking leg quarters 30 to 40 mins. until tender; remove chicken and vegetables. Set broth in pot aside. Discard vegetables. Set chicken aside cool. Remove skin. Pull meat from bones, tearing into large pieces. Set aside, discard bones.
4) For dumplings, turn dough onto well floured surface. Roll very thin about 1/16" cut in 1 1/2 by 2 1/2" pieces. Return broth to boiling. Season well with additional kosher salt. Add dumplings to broth shaking pot occasionally. Do not stir. Cook 3 to 5 mins. Add reserved chicken and egg. Reduce to simmer. Add butter,cream,and few grindings of black pepper. Cook 2 mins. Remove from heat; cover Let stand 10 mins.
