In a large pot over medium heat, melt butter. Add carrots and celery and season mixture with salt and pepper. Cook, stirring often, until the vegetables are tender, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds.

Add flour and stir until the vegetables are coated, then add chicken, thyme, bay leaf, potatoes and chicken broth. Season with salt and pepper. Bring mixture to a simmer and cook for 15 minutes, or until the chicken is no longer pink and the potatoes are tender.

Remove chicken from pot and transfer to a medium bowl. Using two forks, shred the chicken into small pieces and return to the pot. Garnish with parsley.