Heat the oven to 425°F and line an 18-by-13-inch rimmed sheet pan with foil. Toss green beans with 1 tablespoon oil and 1/4 teaspoon salt and spread along the pan edges.
Place flour in one shallow dish. Beat egg and measured water in another, and mix panko, Parmesan, oregano and remaining oil in a third.
Season chicken with remaining salt and pepper. Coat lightly in flour, dip in egg and press into the crumb mixture; arrange in open spaces on the sheet pan.
Roast for 15 minutes, then spoon 2 tablespoons marinara over each cutlet and divide mozzarella across the tops.
Roast for 4 to 6 minutes more until chicken reaches 165°F, cheese melts and green beans are tender. Serve 1 cutlet with one-quarter of the beans per portion.