Preheat oven broiler to 500 degrees, with rack in the highest position.
In a medium bowl, combine tuna, 1/3 cup mayo, onion, capers, lemon juice, salt, pepper, and oregano, and stir until thoroughly combined.
Arrange bread slices in a single layer on baking sheet, and spread the top of each with a little mayonnaise. Divide tuna mixture evenly among slices, then top each with 2 slices of tomato and 1 slice of cheese.
Place baking sheet under the broiler and broil until the cheese is golden brown and bubbling, about 2-4 minutes. Each broiler is different, so you need to WATCH the sandwiches so they do not burn! The cheese will be toasty and melted, but the tuna salad will stay cool and creamy underneath. Makes 4 open-faced sandwiches. Enjoy!