To prepare chicken, heat oil in a large nonstick skillet over medium-high heat. Add ginger and garlic; sauté 15 seconds. Add carrot, zucchini, and bell pepper; sauté 3 minutes or until crisp-tender. Add soy sauce and hoisin sauce; sauté 30 seconds. Place carrot mixture in a bowl; cool 5 minutes. Stir in panko.
Slice each breast half lengthwise, cutting to, but not through, other side. Open halves, laying breast flat. Place each breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/4-inch thickness using a meat mallet or small heavy skillet.
Divide carrot mixture into 4 equal portions; spoon each portion down center of each breast half, leaving a 1/2-inch border at each end. Fold breast sides over filling.
Place a 2-foot-long sheet of heavy-duty plastic wrap on a work surface with 1 long side hanging over the counter's edge 2 inches. Place a stuffed breast half, seam side down, on the end farthest from you; tightly roll the chicken toward you, jelly-roll fashion. Twist the ends in opposite directions to form a cylinder. Tie plastic wrap in tight knots against the chicken on each end. Trim off excess wrap close to the knot. Place a second 2-foot-long sheet of heavy-duty plastic wrap on the work surface; place rolled chicken on wrap, and repeat procedure. Repeat with remaining chicken breast halves.
Bring 3 quarts of water to a boil in a large stockpot; add chicken. Simmer 15 minutes (do not boil), turning occasionally. Remove from water, and let stand 10 minutes before unwrapping and cutting into 1/2-inch-thick slices.
To prepare sauce, combine onions and remaining ingredients in a small bowl. Serve with chicken.