Preheat the air fryer to 375°F (190°C).
In a medium mixing bowl, combine the drained tuna, mayonnaise, diced celery, lemon juice, mustard, and black pepper; mix until the ingredients are fully incorporated.
Apply a thin layer of softened butter to one side of each sourdough bread slice.
Place the bread slices into the air fryer basket, buttered-side down.
Toast the bread at 375°F for 3 minutes until the surface is lightly golden and crisp.
Evenly distribute the tuna mixture onto the toasted side of each bread slice, spreading it to the edges.
Place one slice of sharp cheddar cheese over the tuna layer on each sandwich.
Return the basket to the air fryer and cook at 375°F for 4 to 5 minutes, or until the cheese is completely melted and beginning to brown.
Remove from the air fryer and serve immediately.