In a large bowl, combine all marinade ingredients and stir until well-mixed. Cut chicken breasts into 1-inch pieces and add them to the marinade, stir to coat well. Then cover and refrigerate for 5 to 6 hours.
Thread chicken chunks onto the skewers, leaving about half of each skewer empty for handling. Place them in a single layer on a large baking sheet. Bake at 450 degrees for 10 minutes, then flip the skewers and bake for another 5 minutes or until cooked through. If you'd rather, you can grill the chicken skewers.
While waiting on the chicken to cook, add the peanut sauce ingredients to a small saucepan. Whisk together over medium-low heat until smooth, keep warm over low heat, and stir occasionally.
Transfer chicken skewers to a serving plate. Lightly brush peanut sauce over the chicken. Top with sliced scallions and black pepper; serve while warm.