Cook the vegetables. Melt the butter in a large pot or Dutch oven over medium heat. Then, add the chopped carrots, celery, and onion. Cook for 7-10 minutes, stirring occasionally, until the onion turns translucent and the vegetables are tender. Watch the heat to prevent burning.
Add the flour. Sprinkle the flour over the vegetables, stirring continuously to coat them evenly and break up any clumps. Cook for 2-3 minutes, allowing the flour to combine with the butter and vegetables.
Add the broth and bouillon. Slowly stir in the chicken broth to avoid lumps. Then, increase the heat to medium-high and bring the mixture to a boil. Next, add the chicken bouillon base and ½ teaspoon of black pepper, stirring to blend the flavors.
Cook the chicken. Carefully place the chicken breasts into the pot, fully submerging them in the broth. Cover the pot, reduce the heat to medium-low, and let it simmer for about 15 minutes or until the chicken reaches an internal temperature of 165°F. Then, remove the chicken from the pot and place it on a cutting board. If needed, cook for a few extra minutes until fully done.
Cook the noodles. Add the egg noodles to the pot, stirring to combine. Cover and cook on medium-high for 7-10 minutes until noodles are tender. While the noodles cook, use two forks to shred the chicken on the cutting board into bite-sized pieces. Once the noodles are done, return the shredded chicken to the pot.
Add the cream soup. Whisk together the two cans of cream of chicken soup in a small bowl with ¼ cup of water or milk. Then, pour the mixture into the pot, stirring thoroughly until the soup is creamy and well blended. Taste and add additional pepper and salt if desired.
Garnish and serve. Ladle the soup into bowls and sprinkle with fresh parsley for garnish, if desired. Serve warm.