Season the chicken with 1 tsp salt and the pepper, then brown it skin-side down in the oil in a Dutch oven over medium-high heat, 6 minutes; turn and cook 2 minutes more.
Transfer the chicken to a plate, pour off all but 1 tbsp fat and cook the onion, celery and carrots until the onion softens, 5 minutes.
Return the chicken to the pot with the broth and bay leaf, bring to a gentle simmer and cook partially covered until the chicken reaches 175°F, 30 to 35 minutes.
Remove the chicken, discard its skin and bones and pull the meat into bite-size pieces; discard the bay leaf.
Whisk 1/4 cup flour with 1/2 cup pot broth until smooth, stir it back into the pot and simmer until lightly thickened, 3 minutes.
Whisk the remaining 1 1/2 cups flour, baking powder and remaining 1/2 tsp salt, then rub in the butter until pea-size pieces remain and stir in the buttermilk.
Return the chicken to the simmering broth and drop 12 heaping spoonfuls of dough over the surface with space between them.
Cover tightly and cook without lifting the lid for 10 minutes, then uncover and simmer until the dumplings are dry rather than gummy inside, 5 to 7 minutes; finish with parsley.