Directions

1. Generously season chicken with salt. Cover, and refrigerate overnight.
2. In a large Dutch oven, combine chicken stock, 4 cups water, onion, celery, and a pinch of salt. Bring to a boil over high heat Add the leg portions and the backbone of the chicken to the Dutch oven. Place the breasts, skin side down, on top of the legs. Reduce heat to just below a simmer. Cover partially, and cook, making sure the stock mixture does not come to a simmer, until breasts are cooked through, 20 to 30 minutes. Remove breasts, and set aside. Continue cooking legs until very tender and meat begins to pull away from the bone, 30 to 40 minutes more. Remove chicken and vegetables, reserving liquid. Discard vegetables and set chicken aside until cool enough to handle.
3. Meanwhile, in a medium bowl, whisk together egg, oil, 3 tablespoons cold water, and 1/2 teaspoon salt. Stir in flour until well combined. Cover and refrigerate dough for at least 2 hours or up to overnight.
4. When chicken has cooled, remove skin and discard. Remove meat from bones and tear into 1 1/2- to 2-inch pieces; set aside.
5. Turn dough out onto a generously floured work surface. Roll out dough until it is about 1/16-inch thick. Cut dough into 1 1/2-by-2 1/2-inch rectangles. Bring reserved stock mixture to a boil over high heat, and season with salt. Add dough rectangles to boiling liquid, shaking pot as necessary, but not stirring. Cook until, dough is cooked through, 3 to 5 minutes. Add reserved chicken and reduce to a simmer. Add sliced egg, butter, and heavy cream; season with pepper. Continue cooking 2 to 3 minutes more; season with salt and pepper.
6. Remove from heat and cover; let stand 10 to 15 minutes before serving in shallow bowl.