Set the Instant Pot to SAUTÉ mode. Add olive oil. When hot, add onion, carrots, and celery. Cook until softened, about 5 minutes.
Add chicken, chicken broth, thyme, rosemary, salt, and pepper to the pot. Stir briefly.
Secure the lid and set the valve to SEALING. Select MANUAL or PRESSURE COOK on HIGH for 10 minutes.
When the time is up, allow a Natural Pressure Release (NPR) for 10 minutes, then manually release any remaining pressure.
Remove the chicken and shred it using two forks. Return the shredded chicken to the pot.
In a small bowl, whisk together the flour and cold water to create a slurry. Select SAUTÉ mode again. Stir the slurry into the broth until it begins to thicken.
Stir in the frozen peas and the shredded chicken. Let the mixture simmer for 2 minutes.
Cut the refrigerated biscuit dough into quarters. Drop the dough pieces directly onto the simmering stew, spacing them out slightly. Do not stir them in.
Secure the lid again and set the valve to SEALING. Select MANUAL or PRESSURE COOK on HIGH for 4 minutes.
When the time is up, allow a 5-minute Natural Pressure Release, then quick release any remaining pressure.
Remove the lid carefully. The dumplings should be cooked through. Serve immediately.
Notes
For best results with dumplings, do not overcrowd them; they need space to expand.
If you prefer a thicker stew, increase the slurry amount slightly in step 6.
You can substitute chicken thighs for breasts for richer flavor.