Set the Instant Pot to saute mode and melt the butter. Add the diced onion, celery, and carrots, cooking for 4 minutes until the vegetables begin to soften. Add the garlic and cook for 30 seconds until fragrant.
Pour in the chicken broth and stir in the dried thyme, salt, black pepper, and bay leaf. Nestle the chicken breasts into the liquid.
Lock the lid and set the valve to sealing. Pressure cook on high for 10 minutes.
While the chicken cooks, open the can of biscuit dough and tear each biscuit into 3 to 4 smaller pieces. These will be the dumplings.
When the cook time is complete, do a quick pressure release. Remove the lid, take out the chicken, and shred it with two forks. Discard the bay leaf.
Set the Instant Pot to saute mode. Return the shredded chicken to the pot and pour in the heavy cream. Stir to combine, then drop the biscuit pieces on top of the soup one at a time.
Place the glass lid or a regular lid on top (do not lock the pressure lid) and let the dumplings steam for 10 to 12 minutes until they are fluffy and cooked through.
Gently stir the dumplings into the soup, sprinkle with fresh parsley, and serve immediately.