Step 1

For dumplings: In a medium bowl whisk together egg, the cold water, peanut oil, and 1/2 tsp. salt. Stir in flour; mix until elastic. Cover; chill.

Step 2

Season chicken pieces, including back and neck, with 1 tsp. salt; set aside. In a 6-qt. Dutch oven combine chicken stock, the water, celery, onion, and 1/4 tsp. salt. Bring to boiling. Add in this order: leg quarters, backbone, breast, skin sides down. Reduce heat to below simmer. Cover, leaving a half-inch opening.

Step 3

Cook 30 minutes or until breasts are done (165°F). Remove breasts; set aside. Continue cooking leg quarters and backbone 30 minutes more or until done, (175°F). Remove chicken (keep broth in pot); discard vegetables. When chicken is cool to touch, discard skin and tear meat into pieces; set aside. Discard bones.

Step 4

On a floured surface, use a floured rolling pin to roll dough to 1/16 inch thick. Cut into 1 1/2 × 2 1/2-inch pieces. Return broth to boiling. Season with salt and black pepper. Add dumplings to broth, shaking pot occasionally to prevent sticking. Cook 3 to 5 minutes without stirring. Add reserved chicken; reduce to simmer. Add butter, cream, and additional black pepper. Gently set egg slices on top. Remove from heat; let stand, covered, 10 minutes.