In a mixing bowl, combine soy sauce, fish sauce, brown sugar, minced garlic, ground turmeric, ground coriander, and coconut milk. Add chicken strips and marinate for at least 30 minutes.
Thread the marinated chicken onto skewers.
Preheat the grill to medium-high heat. Grill the chicken skewers for about 10 minutes, turning occasionally, until fully cooked.
For the peanut sauce, combine peanut butter, soy sauce, lime juice, brown sugar, minced garlic, and coconut milk in a bowl. Mix until smooth.
Serve the chicken satay with the peanut sauce on the side.