Combine the tuna (don’t drain) with the drained, chopped artichokes, red peppers, olives, lemon zest, oregano, parsley and salt and pepper to taste. Set aside.
In a separate bowl, combine the roasted garlic with the mayonnaise, Dijon, lemon juice and a little bit of salt and pepper.
Spread the garlic mayonnaise on all four slices of bread, top two slices with provolone, then the tuna-artichoke mixture, then tomato slices and then mozzarella. Top with another slice of rye bread.
Preheat a panini grill, barbeque or cast iron skillet. Brush tops of both sandwiches with olive oil and flipping that side down, place on the grill. Brush what are now the tops of both sandwiches with more olive oil and close grill or wait to flip once bottom is browned. Sandwiches are done when bread is browned and cheese is melted.