Soak the skewers and mix the marinade. Set the bamboo skewers in water. In a medium bowl, whisk the coconut milk, fish sauce, soy sauce, brown sugar, lime juice, turmeric, garlic, ginger, coriander, and pepper until the sugar dissolves.
Cut and marinate the chicken. Pat the thighs dry and cut into long strips about 3/4 inch wide. Add to the marinade, toss to coat, cover, and refrigerate for at least 1 hour and up to 8 hours. Discard leftover marinade after threading.
Make the peanut sauce. Warm the coconut milk in a small saucepan over medium heat until it just simmers. Whisk in the red curry paste, then the peanut butter until smooth. Stir in the fish sauce, brown sugar, lime juice, and soy sauce. Simmer gently 2 to 3 minutes, stirring often, until the sauce coats the back of a spoon. Thin with warm water as needed and adjust lime, sugar, or fish sauce to taste. Keep warm.
Thread the skewers. Thread each chicken strip onto a soaked skewer in a loose S-shape so the meat lies mostly flat, leaving a small gap between folds. You should get about 12 skewers.
Grill the satay. Heat a grill or grill pan to medium-high and oil the grates lightly. Grill 3 to 4 minutes on the first side until caramelized and releasing easily, then flip and grill 3 to 4 minutes more until cooked through to 165°F (74°C) and lightly charred at the edges.
Rest and serve. Rest the skewers 5 minutes. Serve with warm peanut sauce for dipping, lime wedges, cucumber, and steamed jasmine rice if you like.