Preheat a 6-quart air fryer to 370°F for 3 minutes. Cut a horizontal pocket into the thick side of each breast, leaving the opposite edge attached.
Divide mozzarella, sun-dried tomatoes and basil between the pockets and close the openings with wooden toothpicks. Rub the outside with oil, salt, garlic powder and pepper.
Arrange breasts in the basket with space between them. Air fry for 10 minutes, turn gently and cook for 8 to 12 minutes more.
Check that the thickest chicken beside each pocket and the filling center reach 165°F, adding 2-minute intervals as needed.
Rest for 5 minutes, remove every toothpick and slice across the pockets to serve one breast per person.