Heat the olive oil in a large Dutch oven or other heavy pot over medium heat. Pat the chicken dry and season with salt and pepper. Once the pot is hot, add the chicken to the pan and brown on both sides. Then transfer the chicken to a clean plate. (It will not yet be cooked through yet.)
Add the 2 tablespoons butter to the pot. Once melted, add the onion, carrot and celery to the pot and cook until softened, stirring occasionally and scraping up any browned bits from the bottom of the pot, about 5 minutes.
Add the garlic and flour and cook for 30 seconds, stirring.
Stir in the chicken broth, dried thyme, bay leaves, salt and pepper. Return the chicken and any juices from the plate to the pot.
Bring to a low simmer, cover the pot and simmer (adjust the heat as needed to maintain a low simmer) until the chicken is cooked through to 165° F and the vegetables are soft, about 12-16 minutes.
Meanwhile, make the dumpling dough. In a large mixing bowl, whisk together the flour, baking powder, salt and pepper. Pour in the milk and melted butter and stir with a fork until the dough comes together. Place dough in the refrigerator while you finish cooking and shredding the chicken.
Once chicken is cooked through, remove it to a clean bowl or cutting board. Turn the heat off under the pot and let the chicken rest for a few minutes. Then shred the chicken and return it to the pot, along with the frozen peas. Stir. Turn the heat back on under the pot to bring it to a simmer.
Use a spoon to scoop the dumpling dough into 1 ½-inch balls and place on top of the soup in a single layer. Spoon a little of the liquid from the pot over the tops of the dumplings. Once simmering, turn the heat down to a low simmer and cover the pot. Cook at a gentle simmer for 15 minutes, without lifting the lid off of the pot so that the dumplings can steam cook. After 15 minutes split a dumpling in half to see if it is cooked through. If not, cook, covered, for another minute or two. Then remove the pot from the heat.
Discard bay leaves and serve, garnished with chopped fresh parsley if desired.