In a mixing bowl, combine coconut milk, soy sauce, brown sugar, lime juice, minced garlic, grated ginger, red curry paste, salt, and pepper. Whisk until well mixed.
Add chicken pieces to the marinade, ensuring they are well coated. Cover the bowl and refrigerate for at least 15 minutes, up to 2 hours for more flavor.
If using wooden skewers, soak them in water for about 30 minutes to prevent burning during grilling.
Preheat the grill or grill pan to medium-high heat.
Thread the marinated chicken pieces onto the skewers, leaving a small gap between each piece for even cooking.
Grill the skewers for about 4-5 minutes on each side, or until the chicken is cooked through and has nice grill marks. The internal temperature should reach 165°F (75°C).
Once cooked, remove the skewers from the grill and let them rest for a couple of minutes.
Serve the chicken skewers garnished with chopped cilantro and lime wedges on the side.