Make the tonkatsu sauce: In a small bowl combine ketchup, soy sauce, brown sugar, mirin, Worcestershire sauce, grated ginger, and minced garlic; stir until smooth. Set aside and let rest at least 30 minutes.
Pound the chicken: Place each halved chicken breast between plastic wrap and gently pound with a meat mallet or rolling pin until about 1/4 inch thick.
Season and rest: Salt both sides of the chicken liberally, cover, and refrigerate for about 1 hour to improve juiciness (optional but recommended).
Prepare breading station: Put the flour in one shallow bowl, beat the eggs with 1 1/2 tablespoons water in a second bowl, and place the panko in a third bowl.
Bread the chicken: Dredge each piece in flour and shake off excess, dip into the egg wash, then press into panko to coat evenly. Place coated pieces on a wire rack.
Preheat the oven to 200°F and add enough vegetable oil to coat the bottom 1/2 inch of a large skillet; heat oil to about 350°F (medium-high).
Pan-fry the chicken: Working without overcrowding, add chicken to the hot oil and cook 1–2 minutes per side just until the crust sets and turns golden, flipping as needed.
Finish in the oven: Transfer fried chicken to a clean wire rack and keep in the preheated oven to finish cooking, about 7–8 minutes depending on thickness, until internal temperature reaches 165°F.
Slice and serve: When cooked through, cut the chicken into thin strips and serve with rice and the prepared tonkatsu sauce.
