Prepare the Chicken Katsu: Season the chicken breasts with salt and pepper. Set up a breading station: Place flour in one shallow dish, beaten eggs in another, and panko breadcrumbs in a third dish.
Dredge each chicken breast in flour, shaking off excess. Dip into the beaten egg, allowing excess to drip off. Finally, coat evenly in panko breadcrumbs, pressing lightly to adhere.
Fry the Chicken Katsu: Heat 1/2 inch of vegetable oil in a large skillet over medium heat. Once hot, carefully place the breaded chicken breasts in the oil. Fry for 3-4 minutes per side, or until golden brown and cooked through. Transfer to a wire rack to drain excess oil.
Make the Curry Sauce: While the chicken is frying, prepare the curry sauce. In a separate pot or saucepan, heat 1 tablespoon of vegetable oil over medium heat. Add the chopped onion, carrots, and potatoes. Cook for 5-7 minutes, until softened.
Pour in 4 cups of water and bring to a boil. Reduce heat to low, cover, and simmer for 15-20 minutes, or until the vegetables are tender.
Remove the pot from heat and add the curry roux blocks, one at a time, stirring constantly until melted and smooth.
Assemble and Serve: Slice the fried chicken katsu into strips. Serve the curry sauce over steamed rice and top with the crispy chicken katsu.