Melt the butter in a large Dutch oven and saute the onions, carrots, and celery until tender, about 6-8 minutes. Season with salt and pepper.
Stir in the garlic and flour until all veggies are well coated with it.
Pour in the chicken broth and add the chicken breasts, potatoes, red bell pepper, fresh thyme and bay leaf. Season with salt and pepper. Bring to a boil, reduce heat and simmer for 30 minutes.
Using two forks, shred the chicken breast and add the green beans. Cook for 10 more minutes.
Remove the bay leaves, and thyme stems. Serve with chopped fresh parsley on top.