Season the cavity of the chicken with salt and pepper, put the chicken in a kettle just large enough to hold it, and add 14 cups water. Bring the water to a boil, simmer the chicken, covered, skimming the froth, for 45 minutes or less, or until it is tender, and transfer it to a large bowl. When the chicken is cool enough to handle, discard the skin and bones, cut the meat into bite-size pieces, and chill it, covered. Measure the stock remaining in the kettle and boil it until it is reduced to about 10 cups. Strain the stock through a sieve into a bowl, let it cool, uncovered, and chill it, covered, overnight.
In a bowl blend together the flour, the shortening, and the salt. In a small bowl whisk together the egg and 2/3 cup cold water, add the mixture until it forms a soft but not sticky dough. Halve the dough, keeping half of it under an inverted bowl, roll out the other half on a heavily floured surface into a 12- by 10-inch rectangle, and cut it with a pastry wheel into 2-inch squares. Transfer the squares to a lightly floured baking sheet and roll and cut out the remaining dough in the same manner.
Discard the fat from the stock, return the stock to the kettle, and bring it to a boil with the chicken, the carrots, the celery, the onion, the potatoes, peeled and sliced thin, and salt and pepper to taste. Add the noodle squares, a few at a time, stirring gently, simmer the mixture, covered, for 20 to 30 minutes, or until the vegetables and noodles are tender, and stir in the parsley.