Turn on the Instant Pot and select 'Sauté' mode. Add olive oil, then sauté onion, carrots, and celery for 5 minutes until softened. Stir in garlic, thyme, rosemary, paprika, salt, and pepper; cook 1 minute more.
Add chicken broth, water, and chicken breasts to the pot. Secure the lid, set the valve to 'Sealing,' and cook on high pressure for 15 minutes. Allow a natural release for 10 minutes, then perform a quick release.
Remove chicken breasts and shred or dice into bite-sized pieces. Return chicken to the pot along with frozen peas. Press 'Cancel' to turn off Sauté mode.
In a bowl, whisk together flour, baking powder, and salt. Stir in milk and melted butter until just combined—do not overmix.
Using a spoon or small cookie scoop, drop dollops of dough into the simmering soup. Close the lid and cook on high pressure for 8 minutes, then allow a quick release.
Carefully remove dumplings with a slotted spoon and place on a plate. Serve soup with dumplings on top, garnished with fresh parsley if desired.
Notes
For best results, do not overmix the dumpling batter to keep them light and fluffy.