Cut off the beet greens and reserve. Scrub the beets. Using a mandoline or knife, cut the beets into paper-thin slices. Place them in a large bowl and toss with the oil, lemon juice, and salt. Set aside.
Trim the stems from the greens and discard. Wash the leaves. Bring the chicken broth to a boil in a skillet. Add the greens and blanch until wilted and cooked through, about 2 minutes. Drain.
Meanwhile, bring the vinegar to a boil in a small saucepan over medium heat. Reduce the heat and simmer until thickened, 3 to 4 minutes (makes about 1/4 cup). Set aside.
Divide the greens among 4 plates. Top them with the fresh beet salad and drizzle with the balsamic reduction. Garnish with Parmesan shavings.