Melt butter in a Dutch oven over medium-high heat. Saute carrots, celery, and onion for 10 minutes. Add 4 cups water, consomme, salt, and pepper. Bring to a boil. Reduce heat to low and cover. Cook until vegetables are tender, about 15 minutes.

Add turkey and beans to the vegetables. Cook for 5 minutes. Mix flour and Worcestershire sauce with remaining water in a bowl until smooth; stir into turkey mixture. Increase heat and bring to a boil. Reduce heat, cover, and simmer soup until thickened, about 5 minutes.

For the dumplings, combine flour, baking powder, and salt in a large bowl. Stir in parsley and poultry seasoning. Combine milk and egg in a separate bowl; stir into flour mixture until just moistened. Drop mixture by tablespoons onto the simmering soup. Cover and simmer until a toothpick inserted into a dumpling comes out clean, about 20 minutes.