Heat a splash of olive oil in a large pot or Dutch oven over medium-high heat. Season the chicken with salt and pepper on each side. Brown in batches, 4-6 minutes per side. Transfer to a plate; reserve the pot.

Add the celery, carrots, onions, thyme, and garlic to the pot and cook, stirring, until the vegetables begin to soften, about 5-7 minutes. Add the chicken, bay leaves, and stock. Bring to a simmer and cook until the chicken is cooked through, about 25-30 minutes. Discard the bay leaves and transfer the chicken to a plate and let cool let cool. Once the chicken is cooled, shred the chicken with 2 forks and return it to the pot, discarding the skin and bones or use for a treat for your dog.

Whisk together 1/2 cup of the flour, 2 cups of the cooking liquid, and salt and pepper to your liking in a small bowl. Slowly whisk the flour mixture into the pot and simmer until slightly thickened, about 8-10 minutes.

To make the dumplings whisk together the remaining 2 cups of flour, baking powder, baking soda, and 1/4 teaspoon each salt and pepper in a medium bowl. Whisk in the butter, buttermilk, and parsley. Reduce heat to low and drop the mixture into the broth in 8 large spoonfuls. Simmer covered until the dumplings are firm, about 12-15 minutes. Serve sprinkled with parsley.