Prepare the Chicken
Heat 1 Tbsp olive oil in a large pot over medium-high heat. Pound the chicken to an even thickness using a meat mallet. Dab the chicken dry with paper towels and season both sides with salt and pepper.
Sear the Chicken
Sear the chicken in the pot until browned on both sides, about 4 minutes per side (it won’t be fully cooked through yet). Transfer the chicken to a plate and set aside.
Cook the Vegetables
Heat the remaining ½ Tbsp olive oil in the same pot. Add the onion, carrot, and celery, and sauté for 5 minutes. Add the garlic and sauté for an additional minute.
Simmer the Soup
Add the chicken broth, thyme, bay leaves, and season with salt and pepper to taste. Return the chicken breasts to the pot. Bring the soup to a simmer.
Cook the Chicken
Reduce the heat to low, cover with a lid, and let the soup simmer until the chicken is cooked through, about 5–10 minutes (depending on the thickness of the chicken).
Make the Creamy Mixture
While the chicken is cooking, melt butter in a medium saucepan over medium heat. Add flour and cook, stirring constantly, for 2–3 minutes until it’s golden. Slowly add the milk, whisking continuously to smooth out any lumps. Stir in the cream and bring the mixture to a simmer, stirring constantly.
Cook the Noodles
Remove the chicken from the soup and let it rest for 5 minutes before shredding into small bite-sized pieces. Meanwhile, add the noodles to the soup and cook until tender, about 6–12 minutes (or according to package instructions).
Combine and Serve
Turn off the heat for the soup. Pour the milk mixture into the soup, stirring well. Return the shredded chicken to the pot and stir in parsley. Let the soup cool slightly before serving.