Layer ingredients in slow cooker: Place chicken breasts or thighs in the bottom of a 6-quart slow cooker. Add diced carrots, celery, onion, and garlic. Pour in chicken broth, cream soup, thyme, parsley, and bay leaf. Stir gently to combine.
Cook chicken: Cover and cook on low for 6–8 hours or on high for 3–4 hours. Chicken should shred easily with a fork and reach an internal temperature of 165 °F (74 °C).
Shred chicken: Remove chicken from the slow cooker and shred with two forks. Return shredded chicken to the pot.
Thicken broth (optional): If the broth is too thin, mix cornstarch with cold water and stir into the soup. Set the slow cooker to high.
Make dumpling batter: In a medium bowl, combine flour, baking powder, and salt. Stir in milk and melted butter until just combined (do not overmix).
Add dumplings: Drop spoonfuls of dumpling batter onto the surface of the simmering soup. Do not submerge. You should get about 12–16 dumplings.
Cook dumplings: Cover and cook on high for 15–20 minutes, or until dumplings are puffed and cooked through. Do not lift the lid during cooking.
Finish and serve: Remove bay leaf. If using, stir in heavy cream or sour cream. Taste and adjust seasoning with salt and pepper. Serve hot.