Set the oven to broil and move the rack to the highest position.
In a medium bowl, combine tuna, mayonnaise, mustard, red onion, capers, cornichons, lemon juice, thyme, salt, and pepper and stir thoroughly until combined. Leave the tuna in slightly large chunks, if possible.
Arrange bread on a baking sheet and spread mayonnaise on each slice. Divide tuna salad evenly among slices, then top each with 2 slices of tomato and 1 slice of cheese.
Broil tuna melts until cheese is golden brown and bubbling, 3 to 4 minutes. Serve warm.