If the char is frozen, thaw at room temperature about 30 minutes before chopping. If it's raw, freeze for 30 minutes first. Chop the char into 1/4-inch cubes and transfer to a medium mixing bowl.
Fold in the panko, scallions, bell pepper, horseradish, egg white, fumi furikake, kosher salt, and pepper.
Divide the mixture into 4 equal portions and shape into 3/4-inch thick patties. Cover and refrigerate 30 minutes.
Heat the oil in a 12-inch cast-iron pan over medium-high heat until it shimmers. Add the burgers to the pan and cook until lightly browned, about 4 minutes. Flip and cook until lightly browned and the burger has reached an internal temperature of 135°F, another 3 to 4 minutes.
Move the burgers to a plate and rest for 3 minutes before serving on an onion roll with Kewpie mayonnaise.