For the sauce, stir together the Worcestershire sauce, ketchup, and soy sauce, and a pinch of pepper to taste. Set aside.
Heat oil in deep-fryer to 350 degrees F (175 degrees C).
Place flour and panko bread crumbs onto separate plates and season with salt and pepper. Place the beaten egg in a medium bowl. Dip flattened chicken pieces first into flour, then egg, and lastly bread crumbs.
Fry breaded chicken breasts in preheated oil until golden brown and no longer pink in center, about 8 minutes. Transfer to a paper towel-lined plate to absorb excess oil. Slice chicken into thin strips and top with a drizzle of sauce and a sprinkling of sliced green onions. Serve remaining sauce on the side for dipping.