In a bowl, mix white miso and mirin. Texture should be smooth. Spread the mixture on both sides of each fish filet and wrap in plastic wrap. Let sit overnight or up to two days.
When you are ready to serve, bake the filets in the oven at 450 degrees F. The fish is ready when flaky (about 15 minutes). Remove the meat from the skin and remove any bones, then distribute equally among the butter lettuce cups. Garnish with pickled ginger.