In a saucepan, bring the ketchup, applesauce, worcestershire sauce, soy sauce, dijon mustard and rice vinegar to a simmer, transfer to a bowl and let cool.
Put the flour, egg whites and panko in 3 separate bowls. Dredge the pork in the flour, then the egg white, followed by the panko, pressing the crumbs to adhere.
In a large skillet, heat 2 T oil over moderate heat. Add the cutlets and cook until golden brown, about 5 minutes. Brush the remaining oil on the cutlets, flip and cook until golden and cooked through, about 5 minutes longer.
Transfer the tonkatsu to a work surface and cut into strips. Serve with rice and the dipping sauce.
NOTE: The sauce can be refrigerated for up to a month.