Prepare Tuna Mixture: Mix finely chopped red onion, celery, mayonnaise, dill, capers or jalapeños, and lemon juice in a medium bowl. Add oil-packed tuna and mix, breaking it up with a fork. Season with pepper and salt to taste, adding brine if needed.
Butter the Bread: Spread butter on one side of each bread slice. Place slices buttered side down.
Assemble Sandwiches: Place cheese slices on 4 bread slices. Evenly divide the tuna mixture over the cheese. Top with remaining bread slices, buttered side up.
Heat Skillet: Preheat a large cast-iron skillet over medium heat.
Cook Sandwiches: Place sandwiches, cheese side down, in the skillet. Cook, turning once and pressing down with a spatula, until bread is golden brown and cheese is melted, about 4 minutes per side. If using cheddar, cover the skillet after turning to ensure the cheese melts.
Serve: Transfer sandwiches to a cutting board and slice in half. Serve with dill pickles and kettle-style potato chips drizzled with hot sauce.