In a medium bowl, combine the drained tuna, mayonnaise, chopped red onion, chopped celery, lemon juice, salt, and black pepper. Mix well until all ingredients are evenly incorporated.
Preheat a skillet or griddle over medium heat.
Spread the softened butter evenly on one side of each bread slice.
Place two slices of bread, buttered side down, on the skillet.
Divide the tuna salad evenly over the two slices of bread in the skillet.
Place one slice of cheddar cheese on top of the tuna salad on each sandwich.
Top each sandwich with the remaining bread slices, buttered side up.
Cook for about 4-5 minutes until the bottom bread slices are golden brown and the cheese begins to melt.
Carefully flip the sandwiches and cook for another 4-5 minutes until the other side is golden brown and the cheese is fully melted.
Remove the sandwiches from the skillet and let them cool for 1-2 minutes before slicing in half and serving.