Combine all the ingredients in a bowl for the base and set aside.
Make 1/4-inch-deep slashes in the chicken breasts.
In a bowl, combine 2 tablespoons of the ginger-miso base with 2 tablespoons of vegetable oil and spread it all over the chicken. Marinate in the refrigerator for 1 – 6 hours. If you do not have the time, you can skip the marinating step.
Heat grill or grill pan and cook chicken over high heat until cooked through, 7 to 8 minutes. Let cool, then cut the chicken into thin strips.
In a large bowl, whisk 2 tablespoons of base with the lime juice and the remaining 2 tablespoons of oil. Season with salt to taste. Add coleslaw mix and toss well.
Any remaining base you can store in the refrigerator for up to one week. If you like more dressing on your salad, for every one tablespoon of base, add one tablespoon of vegetable oil and a squeeze of lime.
Divide coleslaw on two serving dishes, top each dish with sliced grilled chicken, wonton chips, sliced green onions and serve.