Soak six, 6-inch (15.24cm) wooden skewers in water for 1 hour.
Slice the chicken thinly, across the grain, to make 6 strips.
Thread onto skewers and place in a zip-top bag.
In a cup, whisk 2 tablespoons coconut milk, 1 tablespoon lime juice and 1 teaspoon tamari together, then pour over the chicken.
Seal and gently massage the bag to coat the chicken with marinade. Refrigerate for at least an hour and up to a day.
In a cup, stir the remaining 2 tablespoons of coconut milk, 2 tablespoons lime, 2 teaspoons tamari, honey and sesame oil.
Combine the cabbage, carrots and scallions in a large bowl and drizzle with the coconut milk mixture, toss to coat.
Line a rimmed sheet pan with foil and preheat the broiler.
When the broiler is hot, place the chicken skewers on the pan, not touching, and drizzle any remaining marinade over the chicken.
Broil 6 inches from the heat for about 4 minutes per side.
Serve hot atop the slaw.
