Put the flour in a shallow bowl and dredge the chicken pieces until coated.

In a large stockpot over medium-high heat, melt the butter. Add the chicken and brown on both sides.

Add the carrots, celery, onion, garlic, stock, water, salt, pepper, rosemary, thyme, sage and parsley to the pot. Cover and bring just to a boil. Reduce the heat to low and simmer until the chicken is tender and starting to fall off the bone, about 1 hour. 

Transfer the chicken to a cutting board. When cool enough to handle, use 2 forks to shred the chicken from the bones, discarding the skin and bones. Return the chicken to the pot and bring to a simmer, then prepare the dumpling batter. 

In a bowl, stir together the flour, baking powder, salt and pepper. Stir in the milk and melted butter to make a moist, stiff batter.  

Slightly increase the heat under the pot to a gentle boil. Drop the batter by tablespoonfuls into the pot, or roll into ping pong-size balls and drop into the pot. Simmer, uncovered, for 10 minutes, then cover the pot and simmer for 10 minutes more. 

Remove the lid and let the chicken and dumplings rest for 10 minutes before serving. Serve in large soup bowls and garnish with chopped parsley.