Combine the mayo with the garlic powder in a small plate and set aside.
Mix 2 tablespoons olive oil, 1 tablespoon honey, ½ teaspoon thyme leafs, and a big pinch of salt. On an oven tray covered with parchment paper, place the cherry tomatoes and coat them with the honey and thyme dressing. Bake for 15 minutes at 350 F/180·C and reserve.
Prepare your breading mix. Combine the panko breading and Parmesan cheese, then place on a large plate. Mix the AP flour with a big pinch of salt and pepper in a medium bowl. In addition, whip 1 egg with a pinch of salt in a bowl.
Season eggplant slices with 1 tablespoon olive oil, salt, and pepper on both sides.
Cover eggplant slices with flour, shake off excess. Coat with egg mixture. Finally, cover with the panko and Parmesan breading. Press the slice against the breading well, make sure it’s coated all the way around.
Deep fry on a cast iron skillet with a neutral oil. The oil should cover at least 3" high of the skillet. Each side should take about 1 minute and a half. Once golden, transfer to a wire rack.
Cut the bread roll open. Spread reserved garlic mayo on both slices. Spread olive tapenade on the top slice. Place the baked cherry tomatoes on the bottom slice. Top with the fried eggplant and arugula leafs, then close up the sandwich. Enjoy!