Make dressing: Add rice vinegar, vegetable oil, salt, and ginger to a blender.
Start on low and increase speed until well blended.
Add green onions and stir with a spoon.
Pour half of the dressing into a medium mixing bowl and the balance into a small jar.
Set aside.
Add chicken to the mixing bowl and let marinate in dressing for 2.
In a large skillet over medium heat, add chicken and marinade.
Stir occasionally until chicken is cooked through, 5 to 7.
In a large serving bowl, toss carrots, cucumber, bell pepper, and romaine.
Top with warm chicken and sauce, then garnish with green onions and serve with remaining dressing (from the jar).