Preheat the oven to 450 degrees F (230 degrees C).
Set potatoes on a cutting board. Starting from one end, cut slices that are roughly 1/4-inch thick into each potato, stopping about 1/2-inch from the bottom to keep potato slices connected. Transfer prepared potatoes to a baking sheet.
Stir together olive oil, garlic powder, Italian seasoning, and red pepper flakes in a small bowl. Brush olive oil mixture onto the potatoes using a basting brush, making sure to coat the outer sides and in between the slices. Season liberally with sea salt. Set aside remaining oil mixture.
Bake potatoes in the preheated oven for 25 minutes. Remove from the oven and brush once again with remaining oil. By now the potatoes will have opened a bit, allowing for more oil to slip between the slices. Sprinkle a little more sea salt onto each potato, making sure to sprinkle salt in the slices.
Return to the oven and bake until potatoes are cooked through, 25 to 30 more minutes. Remove from the oven and let cool for 5 minutes. Leave oven on.
Leave a few potato slices unstuffed on both ends of each potato, starting somewhere about 1/4 of the way through the potato. Start by slipping a tomato slice in between your first 2 slices of potato, skip 2 slices, and stuff the next slice with mozzarella cheese and a fresh basil leaf. Skip 2 more, add a tomato slice, and continue alternating tomato and mozzarella cheese/basil leaf until you reach the other end of the potato. Continue until all potatoes have been stuffed. Season each potato with sea salt and cracked pepper.
Return potatoes to the oven and bake until mozzarella has melted, about 5 minutes.
Drizzle with balsamic glaze, sprinkle with grated Grana Padano cheese, and garnish with fresh basil.