Trim lemongrass to 5 inches from base; remove and discard bad outer leaves; cut remaining lemongrass into 1/2-inch lengths. In a food processor, process lemongrass to a hairlike texture. Place in a bowl; add remaining ingredients except chicken.
Remove tenderloins (small piece of meat attached to each breast) from chicken. Pound breasts to 1/8 inch thick; add, with tenderloins, to marinade, turning to coat. Marinate, refrigerated, 30 minutes to 8 hours.
Brush a grill or grill pan with oil; heat until medium hot. Grill chicken until cooked through and juices run clear when pierced, 3 to 4 minutes on each side. Serve.