Place the chicken breasts, diced potatoes, and sliced carrots into the slow cooker.
In a medium bowl, whisk together the cream of chicken soup, chicken broth, heavy cream, and fresh parsley. Pour this mixture over the ingredients in the slow cooker.
Cover and cook on LOW for 6-7 hours or HIGH for 3-4 hours, until the poultry is tender and vegetables are soft.
Remove the chicken from the slow cooker, shred it using two forks, and return it to the pot. Stir in the frozen mixed vegetables (peas, green beans, and corn) and cook for an additional 30 minutes.
Bake the refrigerated biscuit dough in a conventional oven according to the package directions.
Serve the creamy stew in bowls, topping each serving with a warm, baked biscuit.