To prepare dressing, combine first 5 ingredients in a small bowl. Gradually drizzle in oil, stirring constantly with a whisk.
To prepare salad, combine romaine and next 4 ingredients (through chicken) in a bowl; toss gently. Place about 2 cups chicken mixture on each of 4 plates. Top each serving with 1 1/2 teaspoons cashews and 2 tablespoons cheese. Drizzle about 4 teaspoons dressing over each serving.