Place the diced yellow onion in an even layer at the bottom of a 6-quart slow cooker.
Lay the chicken breasts over the onions in a single layer.
In a separate bowl, whisk together the cream of chicken soup, cream of celery soup, poultry seasoning, and black pepper until combined.
Spread the soup mixture over the chicken breasts, then pour the chicken broth around the edges of the slow cooker.
Cover and cook on High for 4 hours, or until the chicken is tender and reaches an internal temperature of 165°F (74°C).
Carefully remove the chicken to a plate and shred into bite-sized pieces using two forks. Return the shredded chicken to the slow cooker.
Stir the thawed mixed vegetables into the chicken and sauce mixture.
Separate the biscuit dough and cut each biscuit into 6 small pieces or thin strips. Flatten them slightly to ensure even cooking.
Place the biscuit pieces on top of the bubbling liquid, pressing them down slightly so they are partially submerged but not layered on top of each other.
Cover and cook on High for an additional 60 to 90 minutes. The dumplings are finished when they have doubled in size and are firm and cooked through in the center.
Gently stir the mixture to coat the dumplings in the creamy sauce, garnish with fresh parsley, and let rest for 10 minutes to thicken before serving.