Preparation Bring broth and water to a simmer in a medium saucepan. Add chicken and poach at a bare simmer until just cooked through, about 6 minutes. Transfer chicken with a slotted spoon to a bowl. Reserve 3 1/2 cups poaching liquid. Meanwhile, whisk eggs with 2 tablespoons butter in a small bowl. Stir in flour, dill, and 1/4 teaspoon salt until just combined (do not overmix). Cook mushrooms and garlic in remaining tablespoon butter with 1/8 teaspoon each of salt and pepper in medium saucepan, stirring occasionally, until softened, 3 to 4 minutes. Add reserved poaching liquid and bring to a simmer. Drop batter by tablespoons into simmering liquid, spacing dumplings evenly. Simmer, covered, until dumplings are puffed and cooked through, about 30 minutes. Slice chicken and add to soup, submerging it, then cook until heated through, about 1 minute.
Preparation Bring broth and water to a simmer in a medium saucepan. Add chicken and poach at a bare simmer until just cooked through, about 6 minutes. Transfer chicken with a slotted spoon to a bowl. Reserve 3 1/2 cups poaching liquid. Meanwhile, whisk eggs with 2 tablespoons butter in a small bowl. Stir in flour, dill, and 1/4 teaspoon salt until just combined (do not overmix). Cook mushrooms and garlic in remaining tablespoon butter with 1/8 teaspoon each of salt and pepper in medium saucepan, stirring occasionally, until softened, 3 to 4 minutes. Add reserved poaching liquid and bring to a simmer. Drop batter by tablespoons into simmering liquid, spacing dumplings evenly. Simmer, covered, until dumplings are puffed and cooked through, about 30 minutes. Slice chicken and add to soup, submerging it, then cook until heated through, about 1 minute.