Dice the chicken into bite-sized pieces. Place in a bowl or on a plate and toss with 1/4 teaspoon of the salt and 1/4 teaspoon of the black pepper. Set aside.
In a Dutch oven or similar large, deep pot, heat the oil and butter over medium high. Add the carrots, celery, potato, onion, poultry seasoning, and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Stir and cook until the vegetables are beginning to soften and the onion is turning translucent, about 5 minutes.
Stir in the garlic and cook 30 additional seconds.
Sprinkle the flour over the top. Whisk and cook until the flour is golden brown, about 30 seconds to 1 minute.
Pour in the milk a few splashes at a time, whisking continuously as you go to remove any lumps. Stir in the broth.
Bring the mixture to a boil. Reduce the heat and let simmer steadily for 15 minutes, stirring periodically.
Meanwhile, prepare the dumplings: In a mixing bowl, whisk together the flour, baking powder, salt, and garlic powder. Pour in the milk. With a rubber spatula or wooden spoon, stir to combine. The batter will seem stiff and should be thick enough to scoop. Let rest for at least 5 minutes.
Stir the chicken and peas into the soup pot. Increase the heat to return to a simmer.
With a small scoop or a spoon, portion the dumpling dough by rough 1/2 tablespoons and drop on top of the soup, spreading the dumplings evenly.
Cover and simmer steadily until the dumplings and the chicken are cooked through, about 5 to 10 minutes more. To check for doneness, remove a dumpling and cut it in half to make sure it's no longer raw inside, or poke it with a fork to check the center (it will still be very moist). Remove from the heat and sprinkle fresh parsley or thyme over the top.
To serve, use a big spoon to scoop down through the top of the dumplings and into the soup underneath, ensuring each serving has a portion of both the soup and dumplings. Enjoy hot, breaking the dumplings up into the soup as needed so that you enjoy a bit of both in each bite.
Notes
*TO USE BISQUICK: Mix 1 cup of Bisquick together with 1/3 cup milk of your choice. Once a dough has formed, drop it by the spoonful on top of the soup when the recipe calls for the dumplings to be added. Finish cooking the recipe as directed.
*TO USE CANNED BISCUITS: Quarter about 6 uncooked canned biscuits. When the recipe calls for the dumplings to be added, scatter the biscuit pieces over the top of the soup instead. Finish cooking as directed.
*TO USE CRESCENT ROLLS: To use crescent roll dough, follow the same instructions as the canned biscuit version, cutting your crescent roll dough into quarters. Adapted from my Crockpot Chicken and Dumplings
TO STORE: Refrigerate leftover chicken dumpling soup in an airtight container for up to 3 days.
TO REHEAT: Warm up leftovers on the stovetop over medium-low heat. The microwave can make the dumplings tough.
TO FREEZE: Freeze leftovers in an airtight storage container or zip-top freezer bag for up to 3 months. Thaw overnight in the refrigerator before reheating according to the instructions above.