Place chicken pieces in a large Dutch oven or heavy-bottomed pot. Add chopped carrots, celery, onion, garlic, bay leaves, and dried thyme. Pour in the chicken broth until everything is covered. Season with salt and pepper.
Bring the mixture to a boil over medium-high heat, then reduce to a gentle simmer. Cover and cook for about 45 minutes, or until the chicken is tender enough to fall off the bone.
Remove chicken pieces from the broth and set aside until cool enough to handle. Remove and discard skin and bones, then shred or cut the meat into bite-sized pieces. Return the chicken meat to the pot and bring the broth back to a simmer.
In a large bowl, combine flour, baking powder, and salt. Cut in the cold butter using a pastry blender or your fingers until the mixture resembles coarse crumbs. Stir in milk just until moistened. If using herbs, fold them in gently.
Drop dumpling dough by spoonfuls (about 2 tablespoons each) onto the simmering broth. Cover the pot tightly and cook for 15 minutes without lifting the lid.
Once dumplings are cooked through, sprinkle with fresh parsley, adjust seasoning if needed, and let stand for 5 minutes before serving.