In a large saucepan, melt butter over medium-high. Add onion, celery, carrots and green beans. Cook until onion is translucent, about 4 minutes. Add flour and cook, stirring, 1 minute. Gradually add broth, stirring constantly, then bring to a boil. Reduce heat and simmer 5 minutes. Stir in chicken and season with salt and pepper. Drop heaping spoonfuls of batter on top of chicken mixture. Cover and simmer until dumplings are cooked through, about 12 minutes.serve topped with additional parsley.
Dumplings
Whisk together 1 cup flour, 1 tsp. each baking powder and coarse salt, and 2 tbsp. Chopped fresh parsley. Cut in 2 tbsp. Cold butter. Stir in 1/2 cup whole milk.