Cook the soba noodles according to the package directions and rinse under cold water.
Cut the chicken into 1/4-inch-thick slices.
Heat the soy sauce and brown sugar in a large skillet over medium heat. Working in batches, add the chicken and sauté until cooked through, about 1 1/2 minutes per side. Transfer to a bowl and repeat with the remaining chicken.
Sprinkle the sesame seeds over the chicken and toss to coat.
In a small bowl, whisk together the ginger, vinegar, and oil. Divide the greens, carrots, and onion among individual plates. Top with the soba noodles and chicken. Serve with the vinaigrette.
Tip: To toast sesame seeds, cook them in a dry skillet over moderate heat, stirring constantly, until golden, 2 to 3 minutes. Transfer them immediately to a cool plate.