Whisk coconut milk, 3 tbsp soy sauce, brown sugar, and 1.5 tbsp lime juice in a bowl.
Stir in 2 tsp garlic, 1.5 tsp ginger, curry powder, turmeric, salt, and pepper. Reserve 2 tbsp marinade in a small cup.
Slice chicken into 1-inch-wide strips. Toss with remaining marinade. Chill 20–30 minutes.
If using wooden skewers, soak them in water while the chicken marinates.
Thread chicken onto skewers, letting excess marinade drip off.
Preheat a grill or grill pan to medium-high. Oil the grates lightly with neutral oil.
Grill skewers 5–7 minutes. Flip, brush lightly with reserved marinade, and cook 4–5 minutes more until 165°F.
Make the sauce: whisk peanut butter, warm water, 1 tbsp soy sauce, rice vinegar, 1 tbsp lime juice, chili sauce, 1 tsp garlic, 1 tsp ginger, and fish sauce until smooth. Thin with a splash more water if needed.
Rest skewers 3 minutes. Sprinkle with cilantro and peanuts. Serve with the peanut sauce.