In pot, throw in hen, leeks and half the vegetables. Cover with cold water. Simmer 1½ hours until hen is tender. Remove hen, strain and reserve liquid.
In soup pot, melt 3 tablespoons butter. Sweat celery carrots and onions. Add meat from hen. Lightly dust with flour. Add liquid. Bring to boil. Reduce to simmer until vegetables are tender. Make dumpling batter. Drop dumpling mix by spoonful onto simmering stew. Reduce heat. Cook 10 minutes uncovered. Cook 10 minutes more covered.
In bowl, whisk eggs and milk. Stir in flour and baking powder. Fold in herbs, salt and melted butter. Mix lightly. Let rest 15 minutes. Drop onto stew and cook as above.