Layer the base ingredients: In a large crockpot, place the chicken thighs or breasts, carrots, celery, onion, and garlic.
Pour in broth and seasoning: Add chicken broth, cream of chicken soup, butter, salt, pepper, and parsley. Stir gently to combine.
Cook low and slow: Cover and cook on low for 6 hours or on high for 3 hours, until the chicken is fully cooked and vegetables are tender.
Shred the chicken: Remove the chicken from the pot, shred with two forks, and return to the crockpot.
Add milk and stir: Pour in the milk and stir well. Taste the broth and adjust seasoning if needed.
Top with dumplings: Place biscuit dough pieces evenly on top of the soup. Cover and cook on high for another 1 to 1 ½ hours, until dumplings are cooked through and fluffy.
Garnish and serve: Spoon the soup into bowls and top with fresh parsley or extra black pepper if desired.