Preheat oven to 375 degrees Fahrenheit.
Line a baking sheet with a silicone mat or tin foil and sprinkle with salt to make one thin layer.
Wash the beets and place them on top of the salt bed and sprinkle salt on the tops of them as well.
Place the beets in the oven and cook for 30-40 minutes. When you insert a paring knife into your largest beet, they are done, and you have little to no resistance.
While the beets are cooking, place the stock in a medium saucepan and bring to a boil.
Add the farro and reduce the heat to simmer and cover. Cook for 15-20 minutes until all the liquid has been absorbed. Fluff with a fork and season with salt, 1 teaspoon sugar, and the1 ½ tablespoons olive oil. Reserve.
Peel the beets while they are still hot, and make sure you do not drop them back into the salt bed. Cube the beets and place them in a medium mixing bowl. Discard the salt bed and skins and mix the cubed beets with the red wine vinegar. (Because this is a composed salad, the beets can stay and be stored in the vinegar bath up to one week in the refrigerator).
To compose the salad: Place arugula bed on a plate or bowl, sprinkle with several spoonfuls of farro. Top with brined beets, toasted pine nuts, shaved Parmigiano Reggiano, and a few mint springs. Drizzle a little olive oil over the top of the salad to finish.