Season the chicken thighs on both sides with the salt and pepper. Heat the oil in a cast iron Dutch oven and, when hot, add the chicken thighs, skin side down, and cook until golden brown, about 4 minutes. Turn the chicken thighs over and add the chopped onion, carrot and celery and cook until the thighs are lightly browned on the second side, about 3 minutes. Add the thyme, bay leaf, and chicken broth. Cover the pan and cook on medium heat until the thighs are tender and the meat pulls away from the bone easily, 25 to 30 minutes.
While the chicken is cooking, combine the butter and flour in a small heatproof bowl and, using your fingers or a pastry blender, blend until smooth. Set aside.
Using a slotted spoon, remove the chicken thighs and chopped vegetables from the cooking liquid and transfer to a plate to cool. Ladle 1 cup of the hot cooking liquid into the bowl with the butter-flour mixture and whisk to combine. Add this mixture to the pan along with the onion chunks, sliced carrot and sliced celery and stir to combine. Cover and cook for 10 minutes over medium heat, or until the vegetables are just tender.
While the veggies are cooking, make the dumplings by combining the flour, baking powder and salt in a small mixing bowl. Heat the milk and butter until warm and the butter is melted. Add the warm milk mixture to the flour mixture and stir until just combined. Transfer to a floured work surface and knead lightly once or twice. Using a floured rolling pin, roll to a thickness of 1/8-inch. Using a knife, cut the rolled out dumpling batter into 1 1/2-inch strips.
When the chicken has cooled enough to handle, remove the meat from the bones and discard the bones, chicken skin and cooked vegetables. Reserve the meat separately.
When the veggies are crisp-tender, uncover the pan and add the reserved chicken meat, peas, and parsley to the cooking liquid. Stir to combine, increase the heat to medium-high and, when the mixture comes to a boil, add the dumplings, 1 strip at a time, to the pan, using a spoon to gently submerge the dumplings in the hot mixture. Cook, uncovered and undisturbed, until the dumplings are cooked through, 7 to 10 minutes. Stir gently and serve in wide shallow bowls.