Heat broiler, with rack in highest position. In a medium bowl, combine tuna, mayonnaise, onion, capers, lemon juice, salt, pepper, and oregano and stir until thoroughly combined.
Arrange bread on a baking sheet and spread a little mayonnaise on each slice. Divide tuna salad evenly among slices, then top each with 2 slices tomato and 1 slice cheese. Broil until cheese is golden brown and bubbling, 3 to 4 minutes.