Pat chicken dry and season generously with salt and pepper.
Heat a large Dutch oven over medium-high. Add olive oil and butter.
Sear chicken skin-side down until deeply golden, about 5–7 minutes per side. Work in batches so pieces don’t crowd. Transfer to a plate.
Add onion, celery, and carrots to the pot with a pinch of salt. Cook until softened, 5–7 minutes, scraping up browned bits.
Stir in garlic, thyme, and bay leaves. Cook 1 minute until fragrant.
Return chicken to the pot. Pour in broth, bringing liquid just over the chicken. Bring to a boil, then reduce to a gentle simmer.
Cover and simmer 30–40 minutes, until chicken turns very tender and registers 175°F in the thickest part.
Transfer chicken to a board. Discard skin and bones; shred the meat into bite-size pieces.
Whisk flour with milk until smooth. Bring the pot to a low simmer and slowly whisk in the mixture. Cook 5–8 minutes, stirring, until the broth thickens into a silky gravy. Season with salt and pepper to taste.
Stir shredded chicken back into the pot. Keep at a very gentle simmer.
In a bowl, whisk flour, baking powder, baking soda, and salt. Stir in parsley if using.
Cut in cold butter with a pastry cutter or fingertips until pea-sized crumbs form.
Pour in buttermilk and stir just until a shaggy, sticky dough forms. Do not overmix.
Keep the stew at a steady simmer, not a hard boil.
Drop rounded tablespoonfuls of dough onto the surface, spacing slightly. Cover the pot tightly.
Cook 12 minutes without lifting the lid. Resist peeking—steam cooks the dumplings.
Uncover and check one dumpling. It should look set and fluffy, not doughy inside. If needed, cover and cook 2–3 minutes more.
Taste and adjust seasoning. Finish with a sprinkle of fresh herbs and a few grinds of pepper.
