In a small mixing bowl stir together mayonnaise or salad dressing, Parmesan cheese, lemon juice, garlic, and, if desired, anchovy paste. Set mayonnaise mixture aside.
In a deep 3- or 3-1/2-quart bowl toss together romaine, chicken or turkey, mushrooms, and cherry tomatoes. Carefully spread mayonnaise mixture evenly over top of romaine mixture, sealing to edge of bowl. Cover tightly with plastic wrap. Refrigerate for 4 to 24 hours.
To serve, lightly toss salad mixture until well coated. Spoon mixture onto salad plates. Makes 6 servings.