Cut coriander roots off at the stem, and mince thoroughly. Set aside a few leaves for garnish. In a blender or food processor, combine coriander roots and leaves, garlic, chillis, turmeric, curry powder, sugar and salt. Process to a coarse paste. Pour in fish sauce and blend until smooth.
Place chicken in a large shallow dish. Rub with the coriander paste. Cover and marinate in the refrigerator at least 3 hours or overnight.
Preheat barbecue for high heat.
Lightly oil the cooking grate. Place chicken on the prepared barbecue, and brush liberally with coconut milk. Barbecue chicken 8 to 15 minutes on each side, depending on the size of the pieces. Turn only once, and baste occasionally with coconut cream. Cook until browned and tender, and juices run clear.