To make the mayonnaise, process the egg, egg yolk, Dijon mustard, rice vinegar, liquid shio koji and salt together in a food processor or blender for about 30 seconds. With the machine running, slowly add the oil, being careful not to break the mayonnaise. If it gets too thick, add water 1 tablespoon at a time to loosen it up.
Bring the mirin, vinegar and salt to a boil in a small saucepan, then immediately remove from the heat and allow to cool.
In a small bowl, whisk together 1 cup mayonnaise, the white miso, ponzu and liquid shio koji. This is your Miso Mayo.
Cut the cabbage head in half lengthwise, then cut each half lengthwise into quarters. Shave the cabbage as thinly as possible using a knife or mandolin. Soak the shaved cabbage in ice water for at least 10 minutes; this will give it a wonderful crunch. Drain well, lightly pat dry with towels, transfer to a bowl, and dress with the cooled mirin/vinegar mixture.
Spoon a pile of Miso Mayo on a plate and gently spread it out using the back of the spoon. Place the dressed cabbage on top, and serve.
