Mix together the vinegar, honey, soy sauce and mirin in a small bowl.
Halve the cucumber lengthwise and scoop out the seeds. Push the cucumber, fennel and radishes thru a food processor with a slicing attachment. Otherwise, slice thinly by hand.
Trim any fat from the beef and slice very thinly. Dust with cornstarch.
Heat 2 tbsp of the oil in a wok or large heavy bottomed pan. Add chile, ginger and beef and stir-fry quickly for 1 minute. Remove to a large plate.
Add the green onions to the wok and fry quickly, about 1 minute. Remove to the plate.
Heat a little more oil in the wok and stir-fry half of the shredded vegetables for about 30 seconds. Remove to the plate. Stir-fry the remaining vegetables and water chestnuts and remove to the plate.
Pour the remaining oil into the wok and add the noodles and cilantro. Cook, stirring, for a few seconds to heat through and to break up the noodles. Add all of the ingredients on the plate back into the wok.
Add the vinegar mixture and mix well stir-frying for about 30 seconds, until heated through. Serve immediately.