Adjust oven to middle position and preheat oven to 350°F (175°C). In a small mixing bowl, combine tuna, mayonnaise, panko (if using), red onion, dill, sweet relish, celery, and mustard. Stir, mashing very well with a fork or stiff spatula, until almost no flakes of tuna remain. Season with salt and pepper.
Using a pastry brush, lightly but evenly brush both sides of each slice of bread with oil, clarified butter, or melted butter. In a large cast iron or nonstick skillet, or using a griddle, toast bread slices on first side over medium heat until nicely browned and crisp, about 3 minutes; using a cooking weight or spatula, gently press down on bread slices while toasting to ensure even heating and browning.
Flip bread and lay one slice of cheese on top of each bread slice. Mound tuna salad on top of 2 of the bread slices, spreading it in an even layer. Close sandwiches, placing top bread slices cheese side down, and continue to cook until bottom of sandwich is well toasted; press down gently on sandwich with cooking weights or a spatula to ensure even heating and browning.
Flip tuna melt and cook until second side is well toasted. If cheese at this point isn't melted enough, transfer sandwiches to a baking sheet and heat in oven until cheese is melted, about 3 minutes. Serve.
