Drain the canned tuna thoroughly and place it in a medium mixing bowl.
Add the mayonnaise, celery, red onion, fresh dill, Dijon mustard, chopped dill pickles, kosher salt, and black pepper to the bowl. Mix until well combined.
Heat a large skillet over medium-low heat. Spread the softened butter evenly on one side of each slice of sourdough bread.
Place two slices of bread, butter-side down, in the skillet. Top each slice with a generous scoop of the tuna mixture, spreading it evenly.
Place two slices of sharp cheddar cheese over the tuna on each sandwich, then top with the remaining bread slices, butter-side up.
Grill for 3 to 4 minutes until the bottom bread is golden brown and crispy. Carefully flip the sandwiches with a spatula.
Cook for another 3 to 4 minutes until the second side is golden brown and the cheese is fully melted.
Remove from the skillet, slice in half, and serve immediately.