In a 5- or 6-quart slow cooker, combine potato, carrot, celery, and onion. Place chicken on top of vegetables. Sprinkle with parsley, garlic, salt, rosemary, thyme, and pepper. Pour broth and wine over mixture in cooker.
Cover and cook on low-heat setting for 7 to 8 hours or on high-heat setting for 3 1/2 to 4 hours.
Using a slotted spoon, transfer chicken and vegetables to a serving platter. Cover with foil and keep warm.
For sauce, skim fat from cooking liquid; strain liquid. In a large saucepan, heat butter over medium heat until melted. Stir in flour; cook for 1 minute. Add cooking liquid. Cook and stir until thickened and bubbly. Cook and stir for 1 minute more.
If desired, sprinkle chicken and vegetables with fresh thyme. Serve with sauce.