Snap off and discard the ends of asparagus. Cut of the tips of the spears and reserve for later. Halve remainder of the spears and blend in food processor with garlic, until very finely minced. Add in the olive oil and 1/2 cup cheese and pulse further. Transfer mixture to a bowl, add in lemon zest, season with salt and pepper and set aside.
Bring a large pot of salty water to a boil, add pasta and cook until not quite al dente. Throw in the tips of asparagus spears reserved earlier to the pot and cook another 1 to 2 minutes, until pasta is al dente.
Scoop 1/4 cup pasta water into a large skillet. Drain pasta and asparagus tips and add to skillet along with asparagus mixture and lemon juice. Cook over low heat while stirring for 2-4 minutes. Taste for seasoning. Serve in warm plates with extra cheese on top.