In a large saucepan, melt 3 tablespoons butter over medium-high. Add onion, celery, and carrots and cook until onion is translucent, about 4 minutes. Add 1/4 cup 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 green beans and season with salt and pepper.
Make dumplings: Whisk together 1 cup flour, baking powder, 1 teaspoon coarse salt, and 2 tablespoons parsley. Cut in 2 tablespoons butter. Stir in milk. Drop heaping spoonfuls batter on top of chicken mixture. Cover and simmer until dumplings are cooked through, about 12 minutes. Serve topped with additional chopped parsley.