In a blender or food processor, combine the softened dried chilies, shallots, garlic, ginger, lemongrass, and ground turmeric. Blend into a smooth paste, adding a splash of water if needed to help the blades turn.

Rub 2 tablespoons of the blended spice paste and half of the salt over the chicken quarters, massaging it under the skin as well. Let the chicken marinate for at least 30 minutes at room temperature, or up to overnight in the refrigerator.

Heat the vegetable oil in a saucepan over medium heat, then add the remaining spice paste. Sauté for 5 to 7 minutes until the paste is highly fragrant and the oil begins to separate slightly from the mixture.

Pour the coconut milk, tamarind water, brown sugar, and remaining salt into the saucepan. Lower the heat and let the sauce simmer gently for 10 to 15 minutes, stirring occasionally, until it thickens into a rich, creamy glaze.

Preheat your outdoor grill or a stovetop grill pan over medium-high heat. Lightly oil the grates to prevent the chicken from sticking.

Place the marinated chicken quarters on the grill and cook for 15 to 20 minutes, turning occasionally, until the skin is charred and the meat is mostly cooked through.

During the last 10 minutes of grilling, generously baste the chicken with the thickened coconut sauce every few minutes. Continue grilling until the chicken reaches an internal temperature of 165°F (74°C) and the glaze becomes sticky and caramelized.

Transfer the grilled chicken to a serving platter and pour any remaining warmed coconut sauce over the top. Serve immediately with steamed white rice and fresh cucumber slices for a complete, satisfying meal.