Place chicken and 1 pinch salt in a large pot; cover with water. Bring to a boil, skimming any scum off the surface with a slotted spoon. Add lemon, celery, onion, and carrots. Simmer broth over medium heat until chicken starts to fall off the bone, about 1 hour.

Discard lemon. Remove chicken from the pot; let cool.

Place flour, salt, and pepper in a large bowl. Make a well in the center. Ladle in 2 to 3 scoops of hot broth. Mix in flour with a spoon until dough comes together into a ball.

Turn dough out onto a floured surface. Knead carefully until smooth. Roll out into a large rectangle; cut into small rectangular dumplings.

Drop dumplings one at a time into the simmering broth. Cook, stirring occasionally, until tender, about 30 minutes.