Heat a panini press or large cast-iron skillet. In a medium bowl, combine the garlic, lemon juice, EVOO (pour and count at an even pace to three), thyme and black pepper to taste. Let stand for a few minutes, then add the tuna, red onion and capers. Mash to combine.
Layer the swiss cheese on 4 slices of bread; top with the tuna, tomatoes, greens, provolone and another slice of bread. Press on the panini press (or in the hot skillet with another skillet weighted down on top). When the sandwiches are brown and crisp and the cheese has melted, cut in half and serve.