Mix flour and shortening together in a bowl until it resembles cornmeal.

Slightly whisk milk and egg together in a separate bowl. Add to flour mixture and stir to form a ball of dough. Place onto a floured surface; knead and roll as thin as possible using a floured rolling pin. Let dough stand for 2 hours.

Place chicken into a large pot and add water until just covered. Bring to a boil over high heat, then continue to boil until no longer pink in the center and meat falls off of the bone, about 1 hour.

Remove chicken from the pot. Pull meat from bones.

Strain broth and return to the pot. Add 2 cups water and bouillon cubes. Add shredded chicken, celery, carrots, and onion. Bring to a boil; continue to boil for 15 minutes.