Cook the noodles according to the package directions and rinse under cold water.
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 ½ 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 plates. Top with the soba noodles and chicken. Serve with the vinaigrette.