Wash beet greens and slice into 1 inchwide strips. Wash and peel beets and cut each into 8 wedges.
Cook beets in large pot of boiling salted water until tender, about 10 minutes. Using slotted spoon, transfer beets to medium bowl. Return water to boil. Add pasta to beet liquid and cook until tender but still firm to bite, stirring occasionally. Drain, reserving 1 cup pasta cooking liquid. Return pasta to pot.
Meanwhile, heat heavy large skillet over medium heat. Add pine nuts and stir until lightly toasted, about 3 minutes. Transfer to small bowl.
Add 2 tablespoons oil and onions to same skillet and sauté until beginning to soften and turn golden, about 10 minutes. Reduce heat to medium-low and continue to sauté until onions are tender and browned, about 30 minutes longer.
Add garlic and stir 2 minutes. Scatter beet greens over onions and garlic. Drizzle remaining 2 tablespoons oil over; cover and cook until beet greens are tender, about 5 minutes.
After draining the pasta and returning it to its cooking pot, stir onion-greens mixture and beets into pasta. Add pasta liquid by 1/4 cupfuls to moisten. Season with salt and coarsely ground black pepper. Stir in 1/3 cup Parmesan cheese or serve it on the side. Transfer to serving dish and sprinkle with pine nuts. Serve, passing additional cheese.