Make the sauce: In a small bowl combine 1/2 cup ketchup, 2 tbsp soy sauce, 2 tsp brown sugar, 1 tbsp mirin, 2 tsp Worcestershire sauce, 1 tsp grated ginger, and 1 tsp minced garlic; stir until smooth. Refrigerate at least 30 minutes to meld flavors.
Pound the chicken: Place each halved chicken breast between plastic wrap and gently pound to about 1/4 inch thickness.
Season and rest: Salt both sides of the chicken liberally, cover, and refrigerate about 1 hour if time allows (optional but recommended for juiciness).
Prepare breading station: Put the flour in one bowl; beat 3 eggs with 1 1/2 tablespoons water in a second bowl; place 3 cups panko in a third bowl.
Bread the chicken: Dredge each piece in flour, shaking off excess; dip into the egg wash; then press and coat evenly with panko. Set coated pieces on a wire rack.
Preheat oven to 200°F and heat about 1/2 inch vegetable oil in a large skillet over medium-high until it reaches roughly 350°F (hot but not smoking).
Fry to set crust: Working in batches to avoid crowding, add chicken to the hot oil and cook 1–2 minutes per side just to set and color the crust; repeat until golden brown on both sides.
Finish in oven: Transfer fried chicken to a clean wire rack and place in the 200°F oven for about 7–8 minutes, or until internal temperature reaches 165°F.
Slice and serve: When cooked through, cut the chicken into thin strips and serve warm with rice and the reserved tonkatsu sauce.