Drain the canned tuna well and place it in a medium bowl; use a fork to flake it apart.
Add the chopped onion, chopped celery, minced garlic, chopped parsley, mayonnaise, 2 tablespoons olive oil, and shredded mozzarella to the bowl; stir until evenly combined.
Season the tuna mixture with salt and pepper to taste and stir again.
Heat a nonstick skillet over medium heat and add the remaining 1 tablespoon olive oil.
Butter one side of a bread slice and place it butter side down in the hot skillet.
Spoon about 3 heaping tablespoons of the tuna mixture onto the bread in the skillet, spreading it slightly but keeping a clear edge to prevent overflow.
Butter one side of a second bread slice and place it butter side up on top of the tuna to form a sandwich.
Cook until the underside is golden brown, about 2–4 minutes, then flip and brown the other side until the bread is golden and the cheese is melted.
Remove the sandwich from the skillet and place on a paper-towel-lined plate to drain briefly; repeat with remaining bread and filling.
