Season chicken pieces with salt and pepper on both sides. In a large Dutch oven, melt the butter over medium-high heat. Add the chicken skin-side down and brown for 5-6 minutes until golden. Flip and brown the other side for 3-4 minutes. Remove chicken to a plate and set aside.
Reduce heat to medium and add diced onions, carrots, and celery to the pot. Sauté for 5-7 minutes until vegetables begin to soften. Add minced garlic and cook for another 30 seconds until fragrant. Stir in dried thyme, sage, and rosemary.
Return the browned chicken to the pot and pour in the chicken broth. Add bay leaves and bring to a gentle boil. Reduce heat to maintain a simmer, cover, and cook for 30 minutes or until chicken is tender and reaches 165°F.
While chicken simmers, prepare the dumplings. In a medium bowl, whisk together flour, baking powder, and salt. In a separate bowl, combine milk and melted butter. Pour wet ingredients into dry and stir gently with a fork until just combined. If using herbs, fold them in gently.
Once chicken is cooked, remove pieces to a cutting board. When cool enough to handle, shred the meat, discarding skin and bones. Return shredded chicken to the pot and stir to distribute. Remove and discard bay leaves.
Stir in heavy cream and adjust seasoning with salt and pepper to taste. If broth seems too thin, create a slurry with 1 tablespoon cornstarch and 2 tablespoons cold water, then stir into the simmering broth.
Bring mixture to a gentle simmer. Drop tablespoon-sized dollops of dumpling mixture onto the surface, leaving about an inch between each. You should have enough for 12-14 dumplings. Cover pot tightly, reduce heat to low, and steam for 15 minutes without lifting the lid.
After 15 minutes, test a dumpling by cutting it in half to ensure it's cooked through. Turn off heat, sprinkle with fresh parsley, and let rest for 5 minutes before serving in deep bowls.