In a mixing bowl, blend the drained tuna with mayonnaise, chopped red onion, lemon juice, and oregano until well incorporated. Adjust consistency by adding more mayo if desired. Season the mixture with salt and pepper.
Butter the exterior sides of each bread slice. Spread the tuna filling evenly over one side of each sandwich base. Lay a slice of cheese atop the tuna mixture, followed by optional toppings such as tomato or avocado. Add a second slice of cheese and close the sandwich with the remaining bread.
Place a skillet over medium-low heat. Grill the sandwiches until the bottom side is golden and the cheese begins to soften. Flip and cook the opposite side until the bread is crisped and cheese is fully melted.