Whisk the katsu sauce: Stir the ketchup, Worcestershire sauce, oyster sauce, and brown sugar together in a small bowl until the sugar dissolves. Set it aside. It thickens a little as it sits and tastes better after 10 minutes.

Make even cutlets: Lay each chicken breast flat and slice it in half horizontally to make 4 thin pieces. Put them between two sheets of plastic wrap and pound the thick spots down until each cutlet is an even 1/2 inch. Pat them dry and season both sides with the salt and pepper.

Set up the breading station: Line up three shallow dishes: the flour in the first, the beaten eggs in the second, and the panko in the third.

Bread the cutlets: Coat a cutlet in flour and shake off the excess, dip it in egg and let the extra drip away, then lay it in the panko and press firmly on both sides until there are no bare patches. Set the breaded cutlets on a plate and let them sit 5 minutes so the coating grabs hold.

Heat the oil: Pour the oil into a 10-inch heavy skillet to a depth of about 1/2 inch, which takes roughly the full 3 cups, and heat over medium until it reaches 340-350°F. A few panko crumbs dropped in should sizzle steadily right away without browning instantly.

Fry: Fry 2 cutlets at a time for about 3 minutes per side, until the crust is deep golden and the thickest part reads 165°F on a thermometer. Thinner cutlets get there in closer to 2 minutes a side, so check early. Drain on a wire rack, not paper towels, and let the oil come back up to temperature before the next batch.

Slice and serve: Rest the cutlets 3 minutes, then cut them crosswise into 1-inch strips. Serve over the cooked rice with a pile of shredded cabbage alongside, the katsu sauce spooned over or served on the side, and the sesame seeds scattered on top.