Preheat a 6-quart air fryer to 390°F for 3 minutes. Put flour in one shallow dish, beat egg with water in a second and mix fine breadcrumbs, Parmesan, salt and oregano in a third.
Coat cutlets lightly in flour, egg and crumbs. Dab both sides with oil, dividing the measured amount across all 4 pieces.
Arrange 2 cutlets in the basket without touching. Air fry for 5 minutes, turn and cook for 5 to 7 minutes more until golden and at least 165°F.
Spoon half the marinara over the cooked cutlets and add half the mozzarella. Air fry for 1 to 2 minutes until the cheese melts, then lift to a plate with a wide spatula.
Repeat with remaining cutlets, sauce and cheese. Rest the final batch for 3 minutes and serve one cutlet per person.