Quarter the ginger into chunks for processing. Cut the scallion into 1" lengths. With a vegetable peeler, trim off the lemon zest (just the yellow part). Halve the lemon and squeeze out 2 tablespoons of juice.
Combine the ginger, scallion, lemon zest, and garlic in the bowl of a food processor and chop finely. Add the lemon juice, cilantro, coconut milk, barbecue sauce, soy sauce, brown sugar and salt. Whiz until everything is finely chopped.
Pour the mixture into a container for marinating. Add the chicken and coat well with the marinade. Cover and refrigerate at least 2 hours, preferably overnight.
Prepare a grill for direct cooking.
Drain off excess marinade from the chicken. Discard remaining marinade. Grill the chicken on both sides over direct medium-high heat, flipping halfway through, until the meat is firm and opaque in the center.