Add the tuna, mayonnaise, minced onion, and celery to a small bowl. Stir to combine.
Spread a thin layer of mayonnaise on one side of each of the slices of bread.
Place two slices of bread, mayonnaise side down in a large pan. Top each slice of bread with a slice of cheese, then half the tuna mixture, and top with another slice of cheese. Top with another slice of bread, mayonnaise side up.
Heat over medium heat for 5 minutes, flip, then heat for an additional 3 minutes.