In a mixing bowl, combine the vegetable oil, soy sauce, brown sugar, minced garlic, minced ginger, turmeric powder, chili powder, salt, and pepper. Whisk until well combined.
Add the chicken pieces to the marinade, making sure they are well coated. Cover and refrigerate for at least 30 minutes, allowing the flavors to infuse.
While the chicken is marinating, prepare the peanut sauce. In a saucepan, combine peanut butter, coconut milk, soy sauce, lime juice, brown sugar, and chili paste. Heat gently over medium-low, stirring continuously until the mixture is smooth and well blended. Adjust seasoning as needed.
Preheat the grill or grill pan over medium heat. If using wooden skewers, soak them in water for 20 minutes to prevent burning.
Thread the marinated chicken pieces onto the skewers, leaving a little space between each piece.
Grill the skewers for about 10-15 minutes, turning occasionally, until the chicken is cooked through and has nice grill marks.
Serve the chicken satay hot with the peanut sauce on the side for dipping. Garnish with chopped peanuts and cilantro if desired.