Bring a large pot of well-salted water to a boil over high heat.
Meanwhile, trim, peel, and halve the beets; then slice them into 1/4-inch-thick half-moons. Remove the stems from the greens and discard. Wash the greens, spin them dry, and slice into 1-inch strips.
Heat 2 Tbs. olive oil in a 6-quart Dutch oven over medium heat. Add the onions and prosciutto and cook, stirring often, until the onion is tender and browned, about 10 minutes.Add the garlic and stir for 2 minutes. Push the onion mixture to the sides and pour 2 Tbs. olive oil into the empty space. Add the beets and cook until just tender, 5 to 7 minutes. Stir in the broth, wine, and rosemary and simmer until the liquid evaporates and the beets are completely tender, 10 to 15 minutes.
Meanwhile, boil the farfalle until al dente, about 1 minute less than package directions. Drain the pasta, reserving 1 cup of the cooking water.
Scatter the beet greens over the beet mixture, drizzle with the remaining 2 Tbs. olive oil, cover, and cook until the greens are tender, about 3 minutes. Stir in the pasta and Parmigiano-Reggiano. If needed, add enough of the pasta cooking liquid to moisten. Season to taste with salt and pepper. Serve with more Parmigiano.