Stir tuna, egg, celery, mayonnaise, walnuts, parsley, and dill together in a bowl until evenly combined. Form mixture into two patties.
Melt butter in a skillet over medium heat. Cook patties in hot butter until firm and golden, 2 to 3 minutes; flip and top each patty with Cheddar cheese. Continue cooking until second side is golden brown, 2 to 3 minutes more.
