Bring a large pot of water to a boil. Meanwhile, heat olive oil in a large saute pan over medium-high heat. Add garlic; cook until aromatic, about 2 minutes. Add lemon zest and greens, season with salt and pepper, and cook until just wilted, about 2 minutes. Remove from the heat and reserve.
When water comes to a boil, season with enough salt so it tastes like seawater, add pasta, and stir continuously until water returns to a boil. Cook pasta according to package instructions. Scoop out about 1 cup pasta water; reserve. Drain pasta through a colander, add to greens, return pan to medium heat, and toss continuously until heated through. Add lemon juice and cheese; toss well, adding pasta water as needed to keep the pasta moist.