Heat the olive oil in a large Dutch oven or other heavy pot over medium heat. Pat the chicken dry and season with salt and pepper. Once the pot is hot, add the chicken to the pan and brown on both sides. Then transfer the chicken to a clean plate. (It will not be cooked through yet.)
Add the butter to the pot. Once melted, add the onion, carrot and celery to the pot and cook until softened, stirring occasionally and scraping up any browned bits from the bottom of the pot, about 5 minutes.
Add the garlic and flour and cook for 30 seconds, stirring.
Stir in the chicken broth. Then stir in the potatoes, Italian seasoning, dried thyme, dried rosemary, bay leaves, salt and pepper. Return the chicken and any juices from the plate to the pot. Nestle the chicken down into the broth.
Bring to a low simmer, then cover the pot and simmer (adjust the heat as needed to maintain a low simmer) until the chicken is cooked through to 165° F and the vegetables are soft, about 20-25 minutes.
Once chicken is cooked through, remove it to a clean bowl or cutting board. When chicken is cool enough to handle, shred it and stir it back into the stew.
Remove and discard the bay leaves. Serve stew garnished with chopped fresh parsley if desired.