Add oil, carrots, celery, onion, salt and pepper. Cook, stirring frequently, until onions are translucent, about 3-4 minutes. Add wine, cook 3-4 minutes, or until reduced by about half. Turn off Instant Pot.
Add broth, chicken, thyme, and rosemary. Secure lid and turn valve to "seal." Set Instant Pot on high pressure (or Manual) for 10 minutes. It will take about 15 minutes for it to come to pressure.
Toward the end of the cooking time, make the dough for the dumplings by combining the flour, baking powder, chives, and salt in a medium bowl. Add milk and melted butter; stir gently to combine.
Quick release pressure when timer goes off. Remove chicken to a plate, and discard bay leaf.
Use two forks to shred chicken into bite size pieces. Stir peas, parsley, and shredded chicken into soup.
Combine cream and cornstarch in a small bowl; stir mixture into hot broth.
Add dumpling dough to soup by tablespoon sized scoops (8-10 dumplings). Cover Instant Pot, secure lid and turn valve to “seal.” Set on high pressure (or Manual) for 2 minutes. Quick Release until valve drops.
Taste and adjust seasoning for salt and pepper.
Shortcut: use 1 small can (7.5 oz.) refrigerated buttermilk biscuits in place of dumpling dough. Cut biscuits in quarters, toss lightly with 1/4 cup flour, and drop into soup. Continue to cook dumplings as directed.
Don't have a pressure cooker? Chicken and dumplings can be made on the stovetop. Simmer chicken and vegetables 30 minutes or until tender. Stir in cream and cornstarch mixture, simmer until slightly thickened. Add dumplings and simmer, covered, about 15 minutes, or until dumplings are cooked through.