Heat oven to 425°F; foil-line 18-by-13-inch rimmed sheet. Toss beans with 1 tablespoon oil and 1/4 teaspoon salt; spread along edges.
Put flour in shallow dish, egg beaten with measured water in another, and panko mixed with Parmesan, oregano, and remaining oil in a third.
Season chicken with remaining salt and pepper. Lightly flour, dip in egg, and press into crumbs; arrange in open pan spaces.
Roast 15 minutes; top each cutlet with 2 tablespoons marinara and evenly divided mozzarella.
Roast 4 to 6 minutes more until chicken reaches 165°F, cheese melted, and beans tender. Serve 1 cutlet with one-quarter beans per portion.