Heat the olive oil in a wide skillet with high sides over medium high. Add the carrots, celery, potato, and onion, and cook until the vegetables are beginning to soften and the onion is turning translucent, about 6 to 8 minutes. Stir in the poultry seasoning, 1/2 teaspoon salt, and pepper, and cook 30 additional seconds. Transfer to the bottom of a 6-quart slow cooker. Keep the skillet handy for later (no need to wash it yet).
Pour the chicken broth into the slow cooker, then arrange the chicken breasts in a single layer on top. Sprinkle the chicken breasts with the remaining 1/4 teaspoon kosher salt.
Cover and cook on low for 2 to 3 hours, until the chicken is cooked through and registers 165°F on an instant-read thermometer. Transfer the chicken to a cutting board to cool. Turn off the slow cooker and cover it.
Prepare the dumplings: In a mixing bowl, whisk together the all-purpose flour, white whole wheat 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.
In the same skillet you used to cook the vegetables, melt the butter over medium heat. Once the butter is melted, 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. Bring to a gentle boil, continuing to whisk to smooth out any residual lumps. Let cook and bubble, whisking all the while, until the mixture thickens, about 3 minutes.
Pour the milk mixture into the slow cooker and stir to combine. Stir in the peas.
At this point, the chicken should be cool enough to handle. Dice it into bite-sized pieces, then stir it into the slow cooker.
Turn the slow cooker to HIGH. With a medium-sized scoop or a spoon, scoop the dumpling dough by generous tablespoons and drop it on top of the soup, spreading the scoops evenly; the top of the soup will be mostly covered, but you'll see some peeking through.
Cover the crockpot and cook on HIGH for 1 additional hour, or until the dumplings are cooked through. To check for doneness, remove a dumpling and cut it in half to make sure it's no longer raw inside (it will still be very moist). Sprinkle the 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 so that you enjoy a bit of both in each bite.
Notes
TO STORE: Refrigerate leftovers in an airtight storage container for up to 3 days.
TO REHEAT: Rewarm chicken and dumplings in a Dutch oven on the stovetop over medium-low heat.
TO FREEZE: Freeze leftovers in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.
*TO USE BISQUICK: Mix 2 cups of Bisquick together with 2/3 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 7 to 8 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.