Place the chicken into a large pot. Add chicken broth and water. Make sure the chicken is covered with water. Bring to a boil and reduce heat to simmering. Throughout cooking skim away any foam that forms in your pot. Simmer for 1 1/2 hours until chicken is done and tender; remove chicken from the broth and allow to cool.
Add carrots, celery and onion to broth and simmer until vegetables are softened.
Debone the chicken once it is cool enough to handle. While deboning the chicken discard the undesirable parts (skin, bones, etc). Cut deboned chicken into bite sized pieces.
For the dumplings, mix your flour, salt, egg, butter and milk. Mix until a ball forms. Add more flour, if necessary.
Form your dumplings by rolling out the dough to about 1/8 inch on a cutting board sprinkled with flour. Using a sharp knife or pizza cutter, cut your dough into bite size pieces.
To your soup, stir in cream of chicken soup and chicken. Bring the chicken soup to a boil.
Drop dumplings into boiling broth, ensuring that they submerge. Add all of your dumplings to the broth. Reduce heat to medium-low, cover and cook for 8-10 minutes.