Place drained tuna in a mixing bowl and break up any larger pieces with a fork. Add the mayonnaise and mix well to evenly coat the tuna.
Add the celery, pickle relish, and onion, and mix well. Taste and add salt and pepper as desired.
Butter one side of each piece of bread and set aside.
Heat a sauté pan or griddle over medium. Add bread, butter side down. (The number of sandwiches you can make at once will depend on the size of your cooking surface.)
Scoop a few tablespoons of the tuna mixture onto the bread. Top with sliced tomato, cheese, and a second piece of bread, butter side up.
Cook for 3-5 minutes, until bottom is golden brown. Carefully flip sandwich and cook another 3-5 minutes, until other side is golden.
Repeat with remaining ingredients. Slice sandwiches and serve warm.