In a small bowl, combine mayo and ⅛ cup thinly sliced basil. Season with salt and pepper, mixing to combine. Spread basil mayo on to the insides of both halves of each bagel.
Flip 2 of the bagel halves over and top each with 2 ounces sliced mozzarella, 1 tablespoon sun dried tomatoes, 1 tablespoon sliced basil, and 2 pieces bacon (broken in half if needed). Top each with ¼ cup freshly shredded parmesan. Top with other bagel halves, placing the mayo side up.
In a large skillet, heat the olive oil over medium heat. Add the bagels, pressing down lightly, and cook for 3-5 minutes until golden. Flip and cover with a large pan or pot. Continue cooking until the mozzarella is melted and bottom bagel halves are golden. Serve immediately.