Spread the kosher salt on a cookie sheet large enough to hold the beets and place the unpeeled beets on the salt.
Place in the oven and cook for 45 minutes. Meanwhile, cut off the woody bottom inch of the asparagus and peel the last three inches of the bottom of the stalk and set aside. When the 45 minutes have passed open the oven door and carefully lay the asparagus around the beets on top of the salt. Return to oven and cook 15 minutes more.
Remove from the oven and remove the asparagus to a platter.
Drizzle with 2 tablespoons olive oil and sprinkle with a bit of the salt in the roasting pan. Allow the beets to cool 5 minutes and then peel them.
Cut the peeled beets into 1/4-inch dice and place in a kitchen bowl. Dress the beets with the vinegar and remaining oil and season them with salt and pepper.
Add the frisee to the beet bowl and toss to mix well. Arrange the beet and frisee mixture over the asparagus, shave shards of Parmigiano over and serve.
