Place the chicken breasts, chicken broth, carrots, celery, onion, garlic, thyme, parsley, bay leaf, salt, and pepper into the Instant Pot. Stir to combine.
Secure the lid and set the valve to "Sealing." Cook on high pressure for 10 minutes.
When cooking is complete, carefully quick-release the pressure. Remove the lid and transfer the chicken to a cutting board. Shred it with two forks and set aside.
Remove and discard the bay leaf. Set the Instant Pot to "Sauté" mode and add the butter. Once melted, whisk in the flour to create a roux, cooking for about 1 minute.
Stir in the heavy cream and return the shredded chicken to the pot. Mix everything together well.
Drop the quartered biscuit pieces into the simmering liquid, spacing them out as evenly as possible. Cover with the lid (valve can be open) and let cook on "Sauté" for 8-10 minutes until the dumplings are cooked through and fluffy.
Turn off the Instant Pot, let it sit for 5 minutes to thicken, then serve hot in bowls. This is pure comfort food heaven!