Add mayonnaise, chipotle pepper, and adobo sauce to a mini food processor bowl then process until smooth. Alternatively, chop pepper up very fine then stir with sauce into mayonnaise.
Spread chipotle mayo onto tops and bottoms of buns then layer on shredded chicken, tomato, red onion, mozzarella cheese, and fresh basil. Press in panini press over medium-high heat until cheese is melted and bread is crisp. Serve immediately.