Pierce skin of each leg several times with a fork. Place chicken skin-side-down on work surface. Score thickest parts of meat with a sharp knife; press and spread lightly so the meat is of even thickness.
Mix together ginger juice, 2 tablespoons sake, the sugar, and the soy sauce in a medium bowl. Add chicken, turning to coat evenly. Marinate 20 minutes, turning occasionally.
Cut leek crosswise into 2-inch lengths; cut each piece lengthwise in half and then into needle-fine strips. Soak in the water 5 minutes; drain and reserve.
Drain chicken, reserving marinade; place chicken skin-side-down on cutting board. Cut onions to the same length as long dimension of chicken pieces. Sift a light, even coating of cornstarch over onions and chicken.
Place 3 or 4 otions along length of each chicken piece; roll each jelly-roll fashion around onions. Fasten each roll securely by wrapping with kitchen twine at 1/2-inch intervals.
Heat oil in 10-inch skillet over medium heat. Add chicken rolls, seam-side-down. Cook, turning frequently with tongs, until light brown on all sides. Pour off excess fat.
Add remaining 1/2 cup of sake to chicken. Reduce heat to medium-low; cook, covered, 7 minutes. Add reserved marinade and cook, covered, turning rolls occasionally, until chicken is tender, 10 to 12 minutes. Remove twine from rolls; cut crosswise into 1-inch thick slices. Spoon pan juices over slices. Serve with the shredded raw leek.