Place the chicken breasts (or thighs), chopped onion, chicken broth, cream of chicken soup, cream of celery soup, garlic powder, dried thyme, and black pepper into a 6-quart slow cooker. Stir gently to mix the liquids and spices around the chicken.
Cover and cook on LOW for 6-7 hours or on HIGH for 3-4 hours, until the chicken is tender and cooked through.
Remove the chicken to a plate or cutting board and shred into bite-sized pieces using two forks. Return the shredded chicken to the slow cooker and stir well.
Open the can of biscuit dough and cut each biscuit into 4 quarters. (If using the frozen vegetables, stir them into the pot now).
Place the biscuit pieces evenly over the top of the chicken mixture. Gently press them down so they are slightly submerged but still visible.
Cover the slow cooker and cook on HIGH for an additional 45 to 60 minutes. Do not lift the lid during this time, as the steam is required to cook the dumplings properly.
Check that the dumplings are fluffy and cooked through (not doughy in the center). Stir gently to coat everything in the sauce and serve warm.