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1. For dumplings, in medium bowl whisk together egg, cold water, oil, and the 1/2 teaspoon salt. Stir in flour. Mix until well-blended and elastic. Cover. Refrigerate 2 hours.
2. Season chicken, including back and neck, with the 1 teaspoon salt. Set aside. In 6-quart 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 down, on top. Reduce heat to just below simmer. Cover, leaving half-inch opening.
3. Cook 30 to 45 minutes or until breasts are just done; remove. Continue cooking leg quarters 30 to 40 minutes 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 inch; cut in 1 1/2- by 2 1/2-inch 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 minutes. Add reserved chicken. Reduce to simmer. Add butter, cream, and few grindings of black pepper. Gently lay hard-cooked eggs on top of other ingredients. Cook 2 minutes. Remove from heat; cover. Let stand 10 minutes. Serves 8.