Whisk soy, brown sugar, and garlic together until sugar dissolves. Place chicken breast halves in a shallow dish or ziploc bag and pour marinade over top. Refrigerate at least 30 minutes.
Using a grill pan or frying pan, cook chicken until browned on both sides and cooked through (juices run clear). Let rest at least 5 minutes before slicing. Slice on the bias.
While chicken is cooking, make your dressing by adding all the dressing ingredients to a small bowl and whisking vigorously to combine. Refrigerate until ready to use, stirring again before use.
While chicken is resting, prepare your salad by placing lettuce, green onions, and carrots in salad bowls/plates. Lay sliced chicken over top of salad and drizzle with dressing. Serve promptly.