Snap the tough butt ends off the asparagus or peel to the tender core.
Cut the stmes into 1" pieces.
Reserve the asparagus tips Bring 5-6 qts water to rolling boil, add 2-3 Tbs salt, and cook asparagus stems for 6-8 minutes until soft and totally tender.
Remove stems with slotted spoon, refresh in cold water, and drain Cook the tips in the boiling water for 3-5 minutes until tender, remove with a slotted spoon, refresh in cold wate, and drain.
Reserve asparagus cooking water.
Puree the stems in a food processor with lemon zest, extra virgin olive oil, 1/2 cup asparagus cookign water, and salt and pepper to taste; transfer the sauce to a 3 qt pot.
Return the remaining asparagus cooking water to a rolling boil, add the pasta, and cook until it still offers considerable resistance to the tooth, around three quarters of the recommended cooking time.
Drain, reserving 2 cups of pasta water.
Add pasta, asparagus tips, and 1/2 cup starchy water to asparagus stem puree and cook ina 3 qt pot over highest heat, stirring for 3-5 minutes until pasta is almost cooked and sauce coats pasta.
Add more pasta water, 1/4 cup at a time, if sauce becomes too dry.
Sauce should surround pasta but be slightly liquid since cheese will thicken it.
Add the grated Parmigiano, heat for an additional minute to melt the cheese and serve immediately.