Vegetable oil, as needed for frying
½ cup all-purpose flour
2 large eggs, beaten
1 ½ cups panko bread crumbs
2 boneless, skinless chicken breasts, halved crosswise then pounded 5mm thick
Kosher salt and black pepper
4 cups tightly packed finely shredded green cabbage
Steamed rice and lemon wedges, for serving
Make the Tonkatsu Sauce
Fill a large cast-iron or heavy skillet with 1/3-inch oil. Heat over medium until an instant-read thermometer registers 175 degrees.
Set up your dredging station. Place flour, eggs and bread crumbs in 3 separate wide, shallow bowls or large plates. Season chicken cutlets with salt and pepper. Working with one cutlet at a time, dredge in flour until fully coated, then shake off excess. Dip in egg, coating both sides, let excess drip off, then press into bread crumbs until well coated. Transfer to a clean plate and repeat with remaining 3 cutlets.
Gently lower 2 cutlets into the oil and fry until golden underneath, about 2 minutes. Adjust heat to keep it as close to 175 degrees C. as possible. Turn over and fry until chicken is golden on the second side and cooked through, 2 minutes longer. Transfer to a baking rack to drain, and season with salt. Repeat with remaining 2 cutlets.
Slice cutlets into thick slices and transfer to plates. Divide the cabbage in mounds next to the katsu. Drizzle the katsu with some of the tonkatsu sauce. Serve with small bowls of rice, lemon wedges and extra tonkatsu sauce.