Season the chicken with salt and pepper. In a Dutch oven over medium-high heat, melt the 2tablespoons butter. Add the chicken and cook, stirring often, until golden, about 4 minutes. Add the carrots, onion, and celery, season with salt and pepper, and cook until the vegetables begin to soften and the chicken is opaque throughout, 4–5 minutes. Sprinkle with the 2tablespoons flour and cook, stirring, for about 2 minutes. Pour in the chicken broth, add the peas, and bring to a boil.
Make the dumpling dough
Meanwhile, in a bowl, combine the 2cups flour, the baking powder, and 1teaspoon salt. Using a pastry blender or 2 knives, cut in the ½cup butter until the mixture forms coarse crumbs the size of peas. Add the milk and stir, then knead a few times until a soft dough forms.
Finish the stew
Drop heaping tablespoons of the dough over the top of the boiling stew. Reduce the heat to low, cover the pot, and cook until the dumplings have nearly doubled in size, 7–10 minutes. Sprinkle with the parsley and serve.