Whisk soy sauce, brown sugar, ginger, and garlic in large bowl. DO AHEAD Can be made 2 hours ahead. Let stand at room temperature.
Prepare barbecue (medium-high heat). Add chicken to marinade; toss to coat. Let marinate at room temperature 15 minutes. Brush grill with oil. Grill chicken until cooked through, about 5 minutes per side.
Wrap grilled chicken with the Sticky Rice in Seasoned Nori Wrappers .
