Place the chicken into a large boiling 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.
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.
Now it's time to prepare your dumplings. Preparing the dumplings is easy. Mix your flour, salt, egg, butter and milk in a stand mixer. If you don't have a stand mixer, a mixing bowl and hand mixer will do the job. Continue to mix until a ball forms.
Form your dumplings by rolling out the dough to about 1/8 inch on a cutting board. Use a sharp knife or pizza cutter to cut your dough into bite size pieces.
Add cream of chicken soup stir in well bring the chicken broth to a boil. Drop dumplings into boiling broth, ensuring that they submerge. Add all of your dumplings to the broth. You can keep dumplings from sticking together by adding butter and dunk dumplings one final time. Reduce heat to medium-low and cook for 8 to 10 minutes. Add chicken and simmer until chicken is heated....your dumplings are now ready to serve.