In a medium bowl, beat 2 large eggs with a fork until well blended. Add the 2 cups of flour and 1/2 teaspoon of salt, then mix with a fork until the dough forms shaggy clumps. Drizzle in the 2 tablespoons of water and mix until a stiff, cohesive dough forms.
Turn the dough out onto a lightly floured surface and knead it for 5 to 7 minutes, until it becomes smooth and elastic. Cover the dough ball with a clean kitchen towel and let it rest for 15 minutes.
Using a rolling pin, roll the rested dough out into a large, thin sheet about 1/8-inch thick. Generously flour the surface of the dough, then loosely roll it up into a log. Slice the roll crosswise into 1/4-inch wide strips. Unfurl the strips, sprinkle them with more flour to prevent sticking, and set the fresh noodles aside.
In a large pot or Dutch oven, bring the 8 cups of chicken broth to a boil over high heat. Add the sliced carrots and celery, reduce the heat to medium, and simmer for 10 minutes, until the vegetables begin to soften.
Drop the prepared egg noodles into the simmering broth and cook for 5 to 7 minutes, stirring occasionally, until they are tender but still have a slight chew.
Stir in the 1 cup of shredded chicken and cook for another 2 minutes, just until the chicken is heated through. Turn off the heat and stir in the 1 tablespoon of fresh parsley. Season the soup with salt and black pepper to taste before serving.