Make the sauce: in a small bowl whisk together ketchup, Worcestershire sauce, molasses, soy sauce, 1/2 teaspoon garlic powder, and grated ginger; cover and refrigerate until serving.
Preheat the oven to 425°F (215°C). Place an oven-safe wire rack on a rimmed baking sheet and lightly spray the rack with nonstick spray.
Prepare the chicken: trim any fat from the chicken breasts and slice each breast crosswise to make 4 cutlets if not already cut; place a cutlet between plastic wrap and pound to 1/4–1/2 inch thickness.
Season both sides of each cutlet with kosher salt and black pepper (about 1/2 teaspoon salt and 1/2 teaspoon pepper per cutlet total, as desired).
Set up breading stations: in one shallow dish combine the all-purpose flour, 1/2 teaspoon salt, and the remaining 1 teaspoon garlic powder; in a second dish whisk the eggs with 1 tablespoon water and 1/2 teaspoon salt; place the panko in a third dish.
Bread the chicken: dredge each cutlet in flour and shake off excess, dip into the egg mixture, then press into panko to coat evenly; transfer to the prepared rack and repeat with remaining cutlets.
Spray the breaded cutlets generously with cooking spray so the panko glistens, then bake on the rack for 15–18 minutes until golden and an instant-read thermometer reads 165°F (74°C).
Slice the cooked cutlets into thick pieces and serve over steamed rice with the sauce, shredded cabbage, and green onions alongside.