Rinse chicken under cool water and place it in the slow cooker with the butter, both cans of cream of chicken soup, diced onion and chicken broth. Stir to just combine, cover and turn the slow cooker on low and cook for 4-6 hours, or until chicken is cooked through.
Check after 4 hours, remove the lid and use a wooden spoon to stir and loosen everything from the sides and the bottom of the pot. Use the wooden spoon to break the chicken into chunks or press the chicken against the side of the pot to shred it if you prefer. The chicken should be very tender and come apart easily.
At this point, you will stir in the celery and carrots (or whatever vegetables you like). Replace the lid and allow the vegetables to begin to cook while you prepare the dumplings.
Pop open the cans of those homestyle buttermilk biscuits. Roll each piece of biscuit in flour to coat it completely. The flour will help thicken the stew. Drop the coated biscuits into the slow cooker on top of the stew. Space the dumplings out as evenly as you can. Once all of the dumplings have been placed in the pot, use your wooden spoon to dunk each one into the hot stew and try to situate the dumplings so they are as evenly spaced as possible.
Cover with the lid once again and allow biscuits to cook for about an hour and a half or until they are cooked through.