Place chopped or torn romaine in a large salad bowl. Cut fennel bulb down the center, then again, quartering it. Remove the core with angled cuts on each quarter bulb. Thinly slice quarter bulbs across and add to salad bowl. Combine romaine and fennel with parsley sprigs and red onion. Place one fingertip over the top of balsamic vinegar to control the pour and splash a few tablespoons into the bowl. Drizzle the salad generously with extra-virgin olive oil and toss lightly to coat salad evenly with vinegar and oil. Season with salt and pepper, to taste, and top salads with curls of Asiago or Parmigiano, created by shaving cheese with your knife.