Combine drained tuna and ½ cup white beans in a medium bowl; using a fork, mash until the mixture begins to come together. Add ¼ cup each celery and pickled onion, 3 tablespoons mayonnaise, 2 tablespoons relish, 1 tablespoon each dill and chives, 2 teaspoons capers and ½ teaspoon pepper; fold until well combined.
Spread ½ tablespoon butter on one side of each bread slice. Flip 2 slices and top each with half of the tuna mixture, 1 cheese slice and another bread slice, buttered-side up.
Heat a large skillet over medium heat. Add the sandwiches; cook, undisturbed, until golden brown on both sides and the cheese is melted, about 2 to 3 minutes per side.
