Preheat oven to 350 degrees F. Remove greens from beets. Cut stems in 1/2-inch pieces and tear leaves into 2-inch pieces; set aside. Place whole, unpeeled beets in a 3-quart rectangular baking dish; add water. Cover tightly. Bake for 1 hour or until tender. Set aside until cool enough to handle.
Peel beets and place in a food processor. Add next six ingredients (through pepper). Process until smooth. Add 2 cups of the flour. Pulse until dough just starts to form. Sprinkle 1/4 cup of the remaining flour on work surface. Place dough on top. Sprinkle remaining 1/4 cup flour over dough. Pat dough out into a 9x9-inch square (1/2-inch-thick). Using a pizza cutter or knife, cut dough into 3/4-inch pieces, rolling to round off edges, if desired. Place on parchment-lined baking sheet until ready to boil.
Bring a large pot of salted water to boiling. Cook, in batches, until gnocchi begin to float. Using a slotted spoon, transfer gnocchi to a tray; pat dry. When cooled, toss gnocchi with nonstick cooking spray to keep them from sticking.
Meanwhile, in a large skillet melt butter over medium heat. Add beet green stems, shallots, and garlic. Cook about 3 minutes or until tender, stirring occasionally. Add beet leaves. Cook 1 to 2 minutes or until wilted. Add gnocchi to the skillet; toss to combine. Remove from heat.
Sprinkle gnocchi with additional grated fresh Parmesan cheese and drizzle with balsamic vinegar.