Add the broth, butter, carrots, celery, onion, garlic, poultry seasoning, thyme, salt, pepper, and bay leaf to a 6-quart slow cooker and stir. Nestle the chicken breasts into the liquid.
Cover and cook on low for 6–7 hours or high for 3–4 hours, until the chicken shreds easily.
Remove the chicken to a cutting board, shred with two forks, and return it to the slow cooker. Discard the bay leaf.
Turn the slow cooker to high. Whisk the cornstarch into the cold water until smooth, then stir it into the pot along with the heavy cream and frozen peas.
Flatten each biscuit with your palm. Cut each into 4–5 narrow strips, then cut those in half. Toss the strips in the flour until lightly coated.
Drop the floured strips into the slow cooker one at a time. Gently press them under the surface with a spoon. Do not stir.
Cover and cook on high for 45–60 minutes, until the dumplings are puffed and cooked through in the center.
Stir once gently, taste and adjust salt, and serve topped with fresh parsley.