Mix flour and shortening together in a bowl until it resembles cornmeal.
Slightly beat egg and milk 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 to just cover. Bring to a boil over high heat and continue to boil until chicken is no longer pink in the center and meat falls off of the bone, about 1 hour. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C).
Remove chicken from the pot; strain broth and return to the pot. Add 2 cups water and bouillon. Remove meat from bones and return to broth. Add celery, carrots, and onion. Bring to a boil; continue to boil for 15 minutes.
Meanwhile, cut dough into strips, approximately 1-inch wide. Tear strips and drop into boiling broth. Continue to boil for 10 minutes more. Season with salt and pepper and serve.
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