For the Kare: in a large heavy bottom pot, melt the butter over medium heat until it begins to bubble. Add the flour, curry powder, and chili powder to the butter and whisk to make a roux (paste of flour and butter). Continue whisking for 3 minutes, or until the smell of raw flour dissipates and the spices are fragrant.
While whisking, slowly pour in one cup of the stock at a time, allowing the roux to thicken the stock before adding in the next cup. Increase the heat to medium-high and add the apple sauce and one tablespoon of kosher salt. Reduce the heat to low and let simmer uncovered while you chop your vegetables.
Chop the carrots, onion, and potato into large 1 to 2 inch pieces and add it to the kare. Add the ground coriander, ground cumin, honey, soy sauce, and rice vinegar. Taste the sauce and adjust the seasonings based on your preference. Let the vegetables cook in the kare uncovered on low heat for about 30 minutes or until tender with a bite.
For the Chicken Katsu: slice each chicken breast in half crosswise then cover the cutting board with a large piece of saran wrap and place a chicken breast in the center. Fold the saran wrap over the chicken and pound it with a mallet or a heavy object like a can or rolling pin. Flatten the thickest part of the chicken breast until the entire breast is roughly ¼ inch thick. Repeat for each piece of chicken. Season the chicken with a generous amount of salt and pepper.
To fry the chicken, pour the canola oil into a large pot for deep frying or cast-iron skillet,until the oil reaches 3-4 inches height. Heat the oil on medium heat. In the meantime, set up a dredging station by placing the starch, panko bread crumbs, and eggs each into separate shallow dishes.
Take each chicken breast and dip it into the potato starch to coat and shake off any excess. Dip each side in the egg mixture to coat, letting any excess drip off. Then dip it in the panko breadcrumbs to coat.
Once the oil reaches 350 degrees, carefully place the chicken into the oil, working in batches to avoid overcrowding the pan, and fry on each side for 3-4 minutes, or until golden brown. Transfer to a paper towel or wire rack to drain off any excess oil and sprinkle with flaky sea salt.
Serve alongside white rice and a heaping serving of warm kare.