Special equipment: a pasta machine
For the broth: Add the chicken, carrots, celery, onion, parsley and 14 cups of cold water to a large pot and bring to boil. Reduce to a simmer and cook over medium heat until the liquid is reduced by one-third, about 1 1/2 hours.
Start on the noodles while the broth is simmering: Mound the flour on a clean work surface. Use your hands and make a well in the center of the mound. Add the eggs, oil and 1/2 teaspoon salt to the center and use a fork to whisk the eggs into the flour, slowly working in the rim of the well until all the flour is completely incorporated.
Knead the pasta for about 8 minutes, adding just a bit more flour if the surface is sticky. If the dough feels too dry, add a drop of water as you go. The dough should feel elastic, smooth and a bit sticky. Shape the dough into a ball and wrap in plastic wrap. Let rest for at least 30 minutes and up to 1 hour at room temperature to let the gluten relax so rolling will be easier.
When the broth is reduced, remove from the heat and strain through a fine mesh sieve lined with a paper towel. Return the broth to the pot and reserve the carrots, celery and chicken. When the chicken is cool enough to handle, pick all the meat from the bones, roughly chop and collect in a bowl. Discard the skin and bones.
Bring the broth back up to a simmer and add the reserved carrots and celery.
Set a pasta machine to the widest setting. Divide the dough into 2 pieces. Roll out each piece in the machine from the widest to the thinnest setting. Cut the pasta using the fettuccine setting.
Add the pasta to the simmering soup by handfuls, stirring between each addition so they don¿t stick. Stir in the chicken after all the pasta has been added. Cook until the pasta is cooked through, 3 to 4 minutes. Taste for seasoning and add salt and pepper, if needed.