Preheat the air fryer on Bake at 375°F and set cooking time for 7 minutes. If using a toaster oven–style air fryer, line the cooking pan with parchment paper.
Spread the butter or mayonnaise on one side of each slice of bread. Place the bread, buttered side down, on a plate or on the lined cooking pan.
Place a slice of cheese on 2 pieces of bread. Top with the remaining bread, buttered side up. Sprinkle the parmesan (if using) evenly over the top of each sandwich.
Place the cooking pan in the air fryer, or if using a basket- style air fryer, place the sandwiches in the basket. Bake the sandwiches in the preheated air fryer.
When the cooking program ends, transfer the sandwiches to a cutting board and cut them into 1-inch-wide strips. Serve right away, with soup alongside for dunking.
Notes
To make extra-gooey and -cheesy sandwiches, double the cheese.