Heat butter in a 3-quart cast iron pot over medium heat. When butter foams, add ginger and cook until fragrant, about 1 minute. Add sake and bring to a boil. Add mussels, stir to coat, cover, and steam until mussels open, about 3 to 5 minutes.
Using a slotted spoon, remove mussels to a serving plate. Add miso to sauce and stir until smooth. Remove from heat, stir in watercress, pour over mussels, and serve.