Combine the garlic, ginger, lemongrass paste, brown sugar, turmeric, and fish sauce in a mixing bowl. Place the chicken pieces in the marinade and gently toss until well coated. Cover and let the chicken marinate in the refrigerator for at least 1 hour or overnight.
Thread the chicken pieces onto the skewers. Cook on a grill or grill pan over medium heat, brush with oil to prevent the meat from sticking. Grill the chicken satays for 3 to 5 minutes on each side, until cooked through.
Meanwhile, saute garlic in oil over medium heat until fragrant. Remove from heat.
Transfer the garlic to a bowl and mix in the rest of the peanut sauce ingredients until well combined. Gradually add warm water to the mixture, stirring constantly, until the desired consistency is reached.
Serve with chicken satay and garnish with cilantro leaves and lime wedges.