Whisk together 3 tablespoons ketchup, 3 tablespoons Worcestershire sauce, 1 tablespoon molasses, 1 teaspoon low-sodium soy sauce, 1/2 teaspoon garlic powder, and 1/8 teaspoon grated fresh ginger in a small bowl; 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.
Trim any fat from the chicken breasts. If using whole breasts, cut crosswise to make 4 cutlets. Place each cutlet between plastic wrap and pound to 1/4 to 1/2-inch thickness.
Season both sides of the pounded cutlets with kosher salt and black pepper to taste.
In a shallow dish, combine 1/2 cup all-purpose flour, 1/2 teaspoon salt, and 1 teaspoon garlic powder.
In a second shallow dish, whisk 2 large eggs with 1/2 teaspoon salt and 1 tablespoon water.
Place 1 1/2 cups panko crumbs in a third shallow dish.
Working one cutlet at a time, dredge in the seasoned flour, shake off excess, dip in the egg mixture, then press into the panko so the cutlet is evenly coated. Transfer to the prepared rack and repeat with remaining cutlets.
Generously spray the breaded cutlets with cooking spray so the panko glistens, then bake on the rack for 15–18 minutes until golden and the internal temperature reaches 165°F (74°C).
Slice the cutlets into thick strips and serve over steamed rice with the prepared sauce, shredded cabbage, and green onions alongside.