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. Drizzle remaining 2 tablespoons oil over; cover and cook until beet greens are tender, about 5 minutes.
Meanwhile, 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 cooking liquid and cook until tender but still firm to bite, stirring occasionally. Drain, reserving 1 cup pasta cooking liquid. Return pasta to pot.
Stir onion-greens mixture and beets into pasta. Add pasta cooking liquid by 1/4 cupfuls to moisten. Season with salt and coarsely ground black pepper. Stir in 1/3 cup Parmesan cheese. Divide pasta among shallow bowls. Sprinkle with pine nuts. Serve, passing additional cheese.
Bon Appétit, February 2009, by Jeanne Thiel Kelley