Place the chopped onion in the bottom of a 6-quart slow cooker. Arrange the chicken breasts in a single layer over the onions.
In a medium bowl, whisk together the cream of chicken soup, cream of celery soup, chicken broth, poultry seasoning, and black pepper until smooth.
Pour the soup mixture over the chicken in the slow cooker. Ensure the chicken is mostly covered.
Cover the slow cooker and cook on HIGH for 4 hours or on LOW for 6 to 7 hours.
Carefully remove the chicken breasts and place them on a cutting board. Confirm the internal temperature has reached 165°F (74°C) using a meat thermometer.
Stir the frozen vegetables into the sauce in the slow cooker. Separate the biscuits and cut each one into four equal pieces. Place the biscuit pieces on top of the liquid.
Using two forks, shred the cooked chicken. Return the shredded chicken to the slow cooker, gently tucking it into the sauce around the dumplings.
Replace the lid and cook on HIGH for another 45 to 60 minutes, until the dumplings are puffed and cooked through.
Gently stir the chicken and dumplings to combine. Sprinkle with fresh parsley and serve warm.