Preheat air fryer to 375°. Using a sharp knife, cut a slit, 1/2 in. deep, down center of each chicken breast, being careful not to cut all the way though. In a small bowl, combine olive oil and seasonings; rub over chicken. Fill opening with onion and green pepper.
Working in batches if needed, place chicken in greased air-fryer basket. Cook 6 minutes; top chicken with cheese slices. Continue to cook until a thermometer inserted in chicken reads at least 165°, 6-8 minutes longer. Repeat with remaining chicken. If desired, serve with optional toppings.