Melt butter in a saucepan and add garlic, onions, and apples. Cover and cook for 5 minutes.
Pour over 1 cup of water and bring to a simmer. Stir in banana, honey, turmeric, curry powder, and ketchup.
Add chicken broth and bring sauce to boil. Season with salt and pepper.
Mix cornstarch with remaining cup of water and add to curry sauce. Simmer for 20 minutes, stirring occasionally.
Strain the sauce through a large sieve or colander placed over a bowl, reserving the curry sauce.
While sauce cooks, prepare chicken by making two diagonal slashes across each breast. Coat chicken in flour on both sides.
Dip each chicken portion in beaten egg, then roll in breadcrumbs.
Heat oil in a wok and deep fry chicken two pieces at a time for 3-4 minutes until golden and crisp.
Remove chicken from pan, drain on absorbent paper, and keep warm.
For vegetable curry, melt butter in separate saucepan and add onions, carrots, and potatoes.
Stir and fry for 5-8 minutes over moderate heat until beginning to turn golden.
Pour in prepared katsu curry sauce, bring to boil, and simmer for 4-5 minutes.
Slice chicken breasts into strips and serve with steamed rice and hot vegetable curry sauce.