Whisk oil, both vinegars, sugar and salt in small, deep microwave-safe bowl. Add onion. Loosely cover with plastic wrap; microwave on HIGH 2 minutes. Remove onions to a separate bowl; set both bowls aside. Can be prepared several hours in advance.
Place prepared lettuce in a large salad bowl. If making several hours ahead, place damp paper towels on lettuce and refrigerate. Use a vegetable peeler to cut curls of Parmesan; place on waxed paper. Curls can stand an hour or so at room temperature.
To serve, sprinkle walnuts and onions over romaine. Whisk dressing and add enough to moisten lettuce, about 3 tablespoons. Toss to combine, then top with Parmesan and serve. Refrigerate remaining dressing for up to a week.