For salad, carefully wash parsley leaves to remove any dirt, then dry well, either in a salad spinner or by placing the leaves in a clean, thin dish towel and wringing out moisture. Put parsley in a salad bowl, add 3 oz. parmigiano-reggiano, and toss well. Add sun-dried tomatoes and garlic and toss again, until salad is well mixed.
For dressing, wash basil leaves to remove any dirt or grit, then dry well and put in the bowl of a food processor. Add oil and vinegar and season to taste with salt and pepper. Process until smooth.
Stir shallots and garlic into dressing. Drizzle dressing over salad and toss until greens are evenly coated. Adjust seasonings to taste, then divide between 6 salad plates. Garnish with remaining 1 oz. parmigiano-reggiano.