In a soup kettle, place chicken breasts at the bottom of the pot. Add the water and chicken broth. Simmer till chicken is tender and about to fall off the bone (30-40 minutes). Carefully remove chicken from pot. Remove chicken from bones and discard bones and any skin from the chicken. Return chicken to the kettle.
Add carrots, celery, onion, thyme, 1 teaspoon pepper, and 1 teaspoon salt. Allow the soup to simmer until the veggies are tender.
In a bowl, mix together the flour, egg, milk, ¼ teaspoon salt, and ¼ teaspoon pepper till smooth. While soup lightly simmers (if soup is not hot enough it will be difficult for the dumplings to hold together and cook), add ½ tablespoon sized scoops evenly into the soup. The dumplings will rise as the cook. Be sure that you do not cluster the dumplings into one spot in the kettle. Season with more salt ad pepper, if desired, and serve.