Place the chicken, carrots, celery and onion in a large Dutch oven or stockpot. Add water, vinegar and salt (adding more water, if necessary, to cover chicken). Bring to a boil. Reduce heat; cover and simmer until meat nearly falls from the bones. Remove chicken from broth; allow to cool. Strain broth, discarding vegetables and seasonings. , Remove meat from bones; discard skin and bones. Cut meat into bite-sized pieces; set aside and keep warm. Set aside 1 cup broth; cool to lukewarm., To make dumplings, combine flour and salt. Make a well in flour; add egg. Gradually stir 1/4 cup reserved broth into egg, picking up flour as you go. Continue until flour is used up, adding additional broth as needed until dough is consistency of pie dough. Pour any remaining reserved broth back into stockpot. , Turn dough onto a floured surface; knead in additional flour to make a stiff dough. Let dough rest for 15 minutes. On a floured surface, roll out dough into a 17-in. square. Cut into 1-in. square pieces. Dust with additional flour; let dry for 30-60 minutes. , Bring broth to a boil (you should have about 4 quarts). Drop dumplings into boiling broth. Reduce heat; cover and simmer until a toothpick inserted into center of a dumpling comes out clean (do not lift the cover while simmering), about 10 minutes. Uncover; add reserved chicken. Stir in pepper.