Preheat the broiler.
In a large bowl, combine the tuna, celery, onion, mayonnaise, and pepper. Mix until well combined. Season to taste with salt, if desired. Note: some brands of tuna may already have salt added, so you may not need to add any additional salt.
Lightly toast the English muffins halves and place on a baking sheet (I line the baking sheet with aluminum foil for easy cleanup).
Divide the tuna salad mixture evenly among the English muffin halves.
Top each with three slices of tomato. Lightly season tomatoes with salt and pepper, if desired.
Top each with one slice of cheese. If the slice of cheese is really big, break it into pieces first so that the cheese stays on top of the tuna melt rather than melting down the sides and onto the baking sheet.
Place the tuna melts under the broiler until the cheese is melted.
Best when served immediately.