In a small stainless steel or glass bowl, whisk together the vinegar, mustard, sugar, salt, pepper, and garlic until the sugar and salt are dissolved. While continuing to whisk, add the olive oil in a thin, steady stream until emulsified. Taste and adjust the seasoning, if necessary.
Place the arugula in a large salad bowl and toss with some of the vinaigrette. Divide the salad among serving plates and garnish with shaved Parmesan. Serve immediately.
Cook's Note: This recipe makes approximately 3/4 cup Balsamic Vinaigrette. Any remaining vinaigrette will keep for up to 2 weeks stored in a non-reactive container in the refrigerator.