In a medium bowl, combine the drained tuna, mayonnaise, Dijon mustard, diced celery, diced red onion, lemon juice, salt, and black pepper. Mix well until creamy and evenly combined.
Lay out 4 slices of sourdough bread. Place one slice of sharp cheddar cheese on each, then divide the tuna mixture evenly on top. Spread it to the edges.
Top the tuna with another slice of cheddar cheese, then close each sandwich with the remaining bread slices. Spread the softened butter on the outside of each sandwich, top and bottom.
Heat a large skillet over medium heat. Place the sandwiches in the skillet and cook for 3 to 4 minutes on the first side until the bread is golden brown and crispy.
Carefully flip the sandwiches and cook for another 3 to 4 minutes until the second side is golden and the cheese is fully melted and gooey.
Let the tuna melts rest for 1 minute, then cut in half and serve hot.