Combine the radish, rice vinegar, and black pepper in a small bowl and let pickle while preparing the rest of the dish.
In a large pot, melt the butter, and then add the potatoes, carrots, onion, and garlic. Cook, stirring often, until slightly garlic is fragrant and vegetables are softened, about 5 minutes. Add the water and bring to a boil.
Add the curry brick and stir until dissolved. Add the apple and then reduce to a simmer and cover. Cook for 20-30 minutes, until potatoes are easily pierced with a fork.
Meanwhile, arrange the flour, eggs, and breadcrumbs in three separate dishes. Heat 1⁄2 inch of vegetable oil in a cast iron pan over medium high heat. Test heat with chop stick - when a chopstick inserted into oil bubbles, it is hot enough. Dredge the chicken cutlets in flour, then egg, then breadcrumb.
Fry for 2-3 minutes per side, until golden brown. Remove to a paper towel lined plate and season with salt. Keep warm.
To serve, plate curry over rice and top with a chicken katsu. Garnish with scallions and pickled radish.