Heat 1 Tbsp olive oil in a large pot over medium-high heat. Pound chicken to an even thickness using a meat mallet then dab chicken dry with paper towels and season both sides with salt and pepper.
Sear chicken in pot until browned on both sides, about 4 minutes per side (it won't be fully cooked through at this point). Transfer to a plate.
Heat remaining 1/2 Tbsp olive oil in pot. Add onion, carrot and celery and saute 5 minutes, then add garlic and saute 1 minute longer.
Add chicken broth, thyme, bay leaves and season with salt and pepper to taste. Add chicken breasts and bring soup to a simmer.
Reduce heat to low, cover with lid and allow soup to cook until chicken has cooked through, about 5 to 10 minutes longer (cook time will vary depending on thickness of chicken breasts).
Meanwhile in a medium saucepan melt butter over medium heat, add flour and cook, stirring constantly, 2 to 3 minutes until it's a deeper golden. While whisking slowly add in milk and whisk vigorously to smooth lumps.
Whisk in cream to milk mixture then bring mixture to a simmer, stirring constantly (scrape bottom edges as well).
Remove cooked chicken from soup and allow to rest for 5 minutes, then shred into small bite size pieces. Meanwhile add noodles to soup, cover and cook until tender, about 6 to 12 minutes (or according to time on package).
Turn heat off soup. Pour milk mixture into soup mixture and return shredded chicken to soup. Stir in parsley. Let soup cool slightly before serving.
Making This Soup In Advance?
This soup won't freeze well because it contains dairy which tends to split and curdle when frozen but you can prepare this in advance and store it in the fridge.
Important tips to remember when making this soup in advance are to not add the noodles until you are serving it. If you add them to the soup prior to serving the noodles will swell up and become mushy.