Break the drained tuna into flakes and fold it with the mayonnaise, celery, pickles, Dijon and black pepper.
Lay out 4 bread slices and top each with 1 cheddar slice, one-quarter of the tuna mixture and a second cheddar slice.
Close with the remaining bread and spread the sandwich exteriors with butter.
Heat a covered skillet over medium-low heat and cook the melts for 3 to 4 minutes on the first side.
Turn the sandwiches, cover for 2 minutes to heat the centers to 165°F, then uncover and cook 1 to 2 minutes until the second side is crisp.
Rest each melt for 1 minute before cutting it in half.