Prepare the Tuna Mixture. In a large mixing bowl, combine the chopped red onion, celery, red bell pepper, mayonnaise, cream cheese, dill, and capers. Stir until well combined. Add the drained tuna and use a fork to gently break apart the tuna while mixing everything until evenly combined. Season with freshly ground black pepper to taste.
Assemble the Sandwiches. Butter one side of each slice of bread (the side that will be toasted). Flip them over and place the cheddar cheese on the dry side of one slice of bread and the tuna mixture on the other slice. Put the slices together to form a sandwich, ensuring the buttered sides are on the outside.
Grill the Sandwiches. Preheat a large skillet or griddle over medium-low heat. Place the sandwiches in the skillet and cook for 2–3 minutes per side, pressing gently with a spatula, until the bread is golden brown and the cheese starts to melt. If necessary, cover the pan for a moment to ensure the cheese melts evenly.
Slice and Serve. After grilling the sandwiches to a golden crisp, remove them from the pan and allow to rest briefly. Using a sharp knife, cut each sandwich in half. This will yield four tuna melt sandwiches. Serve warm alongside pickles, potato chips, and a fresh green salad. Enjoy!