With paper towels, pat the chicken thighs dry. Dice into 1-inch pieces. Season with ½ teaspoon of the salt and 1/4 teaspoon pepper.
Prepare the dumpling dough: Melt the butter in a medium bowl and let cool to room temperature. In a separate, large bowl, whisk together the all-purpose flour, wheat flour, baking soda, sugar, and salt. To the bowl with the butter, add the yogurt and milk and stir to combine (the butter will form some small clumps). Whisk in the egg. Pour the wet ingredients into the dry ingredients. With a rubber spatula, stir just until the ingredients are combined and the batter starts to pull away from the sides of the bowl.
Return to the chicken stew: In a 6-quart or larger Instant Pot, add the olive oil and turn to sauté. Add the carrot, celery, onion, poultry seasoning, and remaining ¼ teaspoon salt. Sauté until the vegetables being to take on color, about 5 minutes.
Add the chicken pieces and sauté until the edges start to take on color, about 3 minutes.
Pour in the chicken broth, scraping up any browned bits from the bottom. Stir in the thyme springs and potato.
Turn the Instant Pot off. With a greased tablespoon, scoop a level amount of the batter, then drop it on top of the stew, using your finger or another spoon to push it off (the dumplings will double in size). Repeat with the remaining batter, placing the dumplings snugly beside each other (there will be some overlap).
Pressure cook on HIGH for 3 minutes, then allow the pressure to release naturally for 15 minutes. Gently vent to release any remaining pressure and turn off the Instant Pot. Uncover the Instant Pot and use a fork to gently break apart any dumplings that have stuck together.
With a slotted spoon, transfer the dumplings to a plate (don't worry if some of the vegetables cling to them).
With the back of the spoon, mash the potatoes in the stew to help it thicken. Stir in the peas, then stir in the cornstarch slurry. Turn the Instant Pot to sauté and let cook a few minutes to thicken further. Carefully taste (it's hot!) and adjust the seasoning as desired. Return the dumplings to the Instant Pot. Sprinkle with fresh parsley and enjoy hot.
Notes
*If you're short on time and don't want to make homemade dumplings, you can swap them for refrigerated biscuit dough. Slice 5 of the biscuits into about 8 pieces (roughly the size of 1 tablespoon). Scatter the pieces of dough over the top of the other ingredients in the Instant Pot and cook as directed in the recipe.
TO STORE: Refrigerate leftovers in an airtight storage container for up to 4 days.
TO REHEAT: Rewarm chicken and dumplings in a Dutch oven on the stovetop over medium-low heat or in the microwave.
TO FREEZE: Freeze leftovers in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.