Four 5-ounce skinless cod fillets, patted dry and halved crosswise
4 hamburger buns, split
3 tablespoons unsalted butter, melted
1/4 pound sharp cheddar, shredded
1 large tomato, sliced crosswise 1/4 inch thick
4 iceberg lettuce leaves
Combine all of the ingredients in a small bowl.
Preheat the broiler. In a medium saucepan, heat 2 inches of peanut oil to 350°. Put the flour in a medium bowl. In a large measuring cup, whisk the ice water with the sake and egg yolk. Whisk the liquid mixture into the flour until the batter is just blended but still slightly lumpy.
Dip the fish in the batter and fry in the hot oil until golden, about 3 minutes per side. Drain on a rack set over a baking sheet.
Set the hamburger buns on a rimmed baking sheet and brush with the melted butter. Broil 6 inches from the heat until toasted, about 1 minute. Sprinkle the cheese evenly over the bun tops and broil until melted, about 30 seconds.
On the bun bottoms, layer the fried cod, tomato slices and lettuce leaves. Top with some of the tartar sauce, close the sandwiches and serve the remaining tartar sauce on the side.