How to Make It
Bring chicken, 1 teaspoon salt, pepper, and broth to a boil in a large saucepan over medium. Cover, reduce heat to medium-low, and simmer 15 minutes or until done; remove chicken and reserve broth.
Cool chicken 30 minutes; skin, bone, and shred or chop chicken, discarding skin and bones. Add chicken to broth.
Melt butter in a stockpot over medium-high. Add carrots, celery, and onion; sauté 5 to 7 minutes or until vegetables are softened and lightly browned.
Add thyme, broth mixture, and 1 cup milk. Bring to a light boil; reduce heat, and simmer 10 to 15 minutes.
Meanwhile, combine flour, cornmeal, parsley, baking powder, and remaining 1 1/2 teaspoons salt and 1 1/2 cups milk in a medium bowl until blended.
Drop dumplings, by 1/4 cupfuls, into simmering broth, stirring gently. (Pot will be very full.) Cover and simmer, stirring often, 15 minutes or until dumplings are done, and serve immediately.