Prepare the chicken katsu: Place each chicken breast between two sheets of plastic wrap. Using a meat mallet or rolling pin, pound the chicken to an even 1/2-inch thickness. Season both sides with salt and pepper.

Set up a breading station: Place the flour in a shallow dish, the beaten eggs in a second shallow dish, and the panko breadcrumbs in a third shallow dish.

Dredge each chicken breast: First, coat thoroughly in flour, shaking off any excess. Next, dip into the beaten eggs, ensuring full coverage. Finally, press firmly into the panko breadcrumbs until completely coated.

Start the curry sauce: Heat 1 tablespoon of vegetable oil in a large pot or Dutch oven over medium heat. Add the chopped onion and cook until softened, about 5-7 minutes. Add the carrots and potatoes, and cook for another 3-5 minutes, stirring occasionally.

Simmer the curry: Pour in the water or chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 10-15 minutes, or until the vegetables are tender.

Add the curry roux: Remove the pot from the heat. Break the curry roux blocks into pieces and add them to the pot. Stir until the roux is completely melted and the sauce has thickened. If the sauce is too thick, add a little more water or broth until desired consistency is reached. Keep warm over very low heat.

Fry the chicken katsu: While the curry simmers, heat 3-4 cups of vegetable oil in a large, deep skillet or Dutch oven over medium-high heat to 350-375°F (175-190°C). Carefully place 1-2 breaded chicken breasts into the hot oil (do not overcrowd the pan).

Cook the katsu: Fry for 3-5 minutes per side, or until golden brown and cooked through (internal temperature reaches 165°F / 74°C). Remove the katsu from the oil and place on a wire rack set over paper towels to drain excess oil.

Assemble and serve: Slice each chicken katsu into strips. Serve immediately with a generous portion of hot white rice and a ladleful of the Japanese curry sauce over the katsu and rice.