Heat vegetable oil in a nonstick skillet over medium-high heat. Add mushrooms; sauté 3 minutes. Combine mushrooms, vinegar, and next 8 ingredients (vinegar though garlic) in a bowl. Add chicken; toss to coat. Cover and refrigerate 30 minutes.
Combine napa cabbage, romaine, and red cabbage. Divide cabbage mixture evenly among 6 plates; top each serving with chicken mixture, bell pepper, and sprouts.