Preheat the oven to 375 degrees F.
Place all the large beets on a sheet tray and bake for 40 to 45 minutes or until a fork slides in and out of the beets easily. Add the baby beets to the tray for the last 20 minutes of cooking time. When the beets have cooled, peel them. Slice the large beets into 1/2-inch thick slices. Cut the baby beets in half.
Place the arugula in a bowl and drizzle in about half of the vinegar and olive oil and season generously with salt. Toss the salad gently to incorporate all of the ingredients. TASTE IT! It should taste very flavorful but not soggy. Arrange the arugula on 4 individual salad plates. Toss all of the beets together in the salad bowl and add the remaining oil, vinegar and salt, and toss. Arrange the dressed beets on the arugula and top with the pistachios and shaved pecorino.