Heat the oil in a large soup pot or Dutch oven over medium-high heat. Add in the onion, carrot, and celery. Sauté until softened. Add in the garlic and sauté for 1 minute until fragrant.
Add in the chicken broth and bring to a boil. Lower in the chicken breasts, cover and simmer until cooked through, about 10-15 minutes. Remove the chicken from the pot and set aside.
Lower the soup to a simmer. Add in the pasta and cook until tender.
Meanwhile, heat the butter over medium heat in a small sauté pan. Once melted, whisk in the flour. Continue to cook, whisking constantly until the mixture is golden, about 1 to 2 minutes. Slowly ladle in 1 cup of the broth while whisking continually. Once incorporated, whisk in the milk and cream. Add to the soup pot.
Once cool enough to tough, shred the chicken into bite sizes pieces with two forks. Return the shredded chicken to the pot. Season with salt and pepper to taste. Stir in the parsley. Ladle into bowls and enjoy!
You can also use chicken thighs! Increase the poaching time to 20 minutes, or until the thighs are cooked through. Feel free to choose your favorite pasta shape in this recipe. Great options are egg noodles, rotini, penne, and macaroni.