Combine the tuna, celery, red onion, capers, mayonnaise, Dijon, anchovy paste, lemon juice, pepper, salt, and oregano in a medium bowl and stir until thoroughly combined.
Preheat a non-stick pan over medium heat.
Dress two pieces of the bread with half of the tuna mixture, half of the sliced tomatoes, if using, and half of the cheese. Cover with two additional slices of bread.
Add half of the butter and half of the olive oil to a hot pan. When hot, add 2 sandwiches and cook until browned on both sides, about 2 minutes each side. Repeat with the remaining bread and tuna mixture. Serve hot, with a side salad of hearts of palm, avocado, tomato, and torn fresh basil leaves, if desired.