In a small bowl, whisk together the ketchup, Worcestershire sauce, molasses, soy sauce, 1/2 teaspoon garlic powder, and grated fresh ginger; cover and refrigerate until ready to serve.
Preheat the oven to 425°F (215°C). Place an oven-safe rack on a rimmed baking sheet and lightly spray the rack with nonstick spray.
Trim any excess fat from the chicken breasts; if using whole breasts, cut each crosswise into two cutlets. Place a cutlet between plastic wrap and pound with a meat mallet or rolling pin until 1/4 to 1/2 inch thick.
Season both sides of the pounded chicken with kosher salt and black pepper (about 1/2 teaspoon salt and 1/2 teaspoon pepper per side, or to taste).
Prepare three shallow dishes: mix the flour with 1/2 teaspoon salt and 1 teaspoon garlic powder in the first; whisk the eggs with 1/2 teaspoon salt and 1 tablespoon water in the second; place the panko in the third.
Working one cutlet at a time, dredge the chicken in the flour, shaking off excess; dip into the egg mixture, let excess drip off, then press firmly into the panko to coat evenly. Transfer breaded cutlets to the prepared rack.
Spray the tops of the breaded cutlets generously with cooking spray so the crumbs glisten. Bake for 15–18 minutes until golden and the internal temperature reaches 165°F (74°C).
Slice the cutlets into thick strips. Serve over steamed rice with shredded cabbage and green onions, and drizzle or serve the chilled sauce on the side.
