To make the vinaigrette 
Combine the shallots, vinegar, and salt in a small bowl and set aside for 5 to 10 minutes to marinate the shallots. Add the olive oil in a slow, steady stream, whisking constantly to combine. Use the vinaigrette or transfer it to an airtight container and refrigerate it for up to 2 days. Bring the vinaigrette to room temperature and whisk to recombine the ingredients before using.

To prepare the beets
Adjust the oven rack to the middle position and preheat the oven to 400ºF.

Put the beets in a baking dish in a single layer, drizzle with the olive oil, season with 2 teaspoons of the salt, and toss to coat the beets with the oil, salt, and pepper. Cover the dish with foil and roast the beets for about 1 hour, until they are tender when pierced with a toothpick or fork. Remove the beets from the oven and remove the foil, being careful not to burn yourself from the steam that will arise from the pan. Set the beets aside until they are cool enough to handle. Rub the beets with a clean dish towel to remove their skins and discard the skins. If using large beets, slice them ½-inch thick and cut the beets into roughly 1-inch segments. Cut medium or small beets into 1-inch pieces.

Put the beets in a medium bowl, drizzle with the lemon juice and half of the vinaigrette. Sprinkle with the remaining ¾ teaspoon of salt and toss to coat the beets with the vinaigrette, lemon juice, and salt. Set the beets aside while you prepare the rest of the salad ingredients, or for at least 15 minutes.

Meanwhile, to prepare the radicchio, put the radicchio leaves in a medium bowl and drizzle the remaining half of the vinaigrette and the lemon juice over the radicchio. Sprinkle the salt over the radicchio and toss to coat the radicchio with the vinaigrette, lemon, and salt. Set the radicchio aside to marinate while you prepare the rest of the salad.

To prepare the cheese balls
Adjust the oven rack to the middle position and preheat the oven to 325ºF.

Spread the pistachios on a baking sheet and toast them in the oven for 10 to 12 minutes, until they’re lightly browned and fragrant, shaking the baking sheet and rotating it from front to back halfway through the cooking time so the nuts brown evenly. Remove the pistachios from the oven and set them aside to cool to room temperature. Put the pistachios in a medium bowl, drizzle with the olive oil, sprinkle with the salt and toss to coat the nuts evenly with the olive oil and salt. Transfer the nuts to a cutting board, finely chop them, and return the chopped nuts to the bowl they were tossed in. Finely grate the zest of half of a lemon over the nuts and toss to distribute the zest.

Roll the cheese into 24 ¼-ounce (bite-sized) balls. Working a few at a time, toss the cheese balls into the bowl with the nuts and roll to coat the cheese on all sides with the nuts. Transfer the pistachio covered cheese balls to a plate or a small baking sheet and continue coating the remaining cheese balls with the remaining nuts in the same way.

To serve
Lift one-third of the radicchio leaves out of the bowl and scatter the leaves to cover the bottom of a large platter. Scatter one-third of the marinated beets over the radicchio and nestle 8 of the cheese balls among the radicchio and beets. Repeat, building a second layer with another third of the radicchio, another third of the beets, and 8 more cheese balls. Build a final layer using the remaining radicchio and the remaining beets and nestling the remaining cheese balls among them.