Add onion to a 6-quart slow cooker and top with chicken breasts. Stir both condensed soups, parsley, poultry seasoning, and black pepper together, spread over the chicken, and add broth.
Cover and cook on High for approximately 5 hours, until the chicken reaches 165°F (74°C) and pulls apart easily.
About 60–90 minutes before serving, roll biscuits thin, cut each into four strips, add thawed vegetables, and place biscuit strips over the mixture. Replace the lid quickly.
When dumpling centers are no longer raw, remove and slightly shred the chicken. Return it to the slow cooker, cook 10 minutes more, and serve hot.