Drain all but about 1 tbsp fat from pot. Add in the diced onion, carrots, and celery. Cook, stirring often, until tender, about 5 minutes.
Return chicken to the pot. Add broth. Over high heat, bring to boil. Then, reduce heat to medium low. Cover and simmer until chicken is tender and cooked through, about 45 minutes. Using a slotted spoon, transfer chicken and vegetables to a large bowl. Set aside pot with broth.
When chicken is cool enough to handle, remove and discard bones and skin. Shred the meat into 2 inch pieces.
Combine the wet mixture and the dry ingredients. Stir until just moist. Gather dough into ball and knead until soft on a floured surface. Using your hand, flatten dough into rectangle about 1/2" thick. Cut the dough into 12 equal pieces.
Carefully drop dumplings into the simmering mixture. Simmer until dumplings are puffed and cooked through, about 10-15 minutes.
Return chicken and veggies to mixture. Cook until warm for a few minutes. Serve and enjoy!