In a bowl, combine egg, mustard, garlic, salt, and pepper. Add chicken and stir to coat completely.
Add some of the panko crumbs to a shallow dish (replenish the crumbs as you use them up). Coat chicken in crumbs.
Heat ½ inch oil in a frying pan over medium heat. Add chicken and cook until golden and crisp on each side, flipping in between. Remove to a paper-towel lined plate. Let cool slightly and slice into very thin strips.
In a bowl, whisk together all dressing ingredients.
Assemble the salad. In a large bowl, toss together cabbages and carrots. Toss with dressing. Add in chicken strips and mix to combine.