Preheat a 6-quart air fryer to 390°F for 3 minutes. Put flour in a shallow dish, egg beaten with water in another, and fine breadcrumbs mixed with Parmesan, salt, and oregano in a third.
Lightly coat cutlets in flour, egg, then crumbs. Dab both sides with oil, dividing measured amount among 4 pieces.
Space 2 cutlets in basket. Air fry 5 minutes; turn and cook 5 to 7 minutes more until golden and at least 165°F.
Top cooked cutlets with half the marinara and mozzarella. Air fry 1 to 2 minutes until cheese melts; lift onto plate with wide spatula.
Repeat remaining cutlets, sauce, and cheese. Rest final batch 3 minutes; serve one cutlet per person.