In a large bowl, combine coconut milk, soy sauce, brown sugar, lime juice, garlic, ginger, red curry paste, salt, and black pepper. Whisk until well blended.
Add the chicken pieces to the marinade, ensuring they are fully coated. Cover and refrigerate for at least 1 hour, or up to 4 hours for more flavor.
Preheat the grill to medium-high heat.
Thread the marinated chicken onto skewers, leaving a little space between each piece for even cooking.
Place the skewers on the grill and cook for 5-7 minutes on each side, or until the chicken is cooked through and has nice grill marks.
Remove from the grill and let rest for a few minutes before serving.