In bowl, whisk together the dressing ingredients (oil, rice vinegar, lime juice, soy sauce, mayonnaise, brown sugar, and garlic). Set aside (for best flavor, refrigerate overnight before using).
In large bowl, gently combine all chopped salad ingredients (cabbage, romaine lettuce, carrots, mandarin oranges, cucumber, green onions, and sesame seeds). Carefully toss the chopped salad with mixed dressing (for best appearance try to preserve the mandarin oranges' shapes). Serve chilled.