Fill a large pot with lightly salted water and bring to a rolling boil. Ease soba noodles into the water and return to a boil. Cook uncovered, stirring occasionally, until noodles are tender yet firm to the bite, 5 to 7 minutes. Drain and rinse under cold water.
Heat soy sauce and brown sugar in a large skillet over medium heat. Add chicken; stir and cook until cubes are no longer pink in the center and juices run clear, 5 to 7 minutes. Transfer to a bowl; sprinkle with sesame seeds, toss to coat.
Whisk olive oil, vinegar, and ginger together in a bowl until vinaigrette is combined. Divide greens among 4 plates; top with soba noodles and chicken. Serve with vinaigrette.