Cook the pasta in salted water according to package instructions. Reserve 1/4 cup of pasta cooking liquid.
In a large pot boil 4 cups water with salt.
When boiling, add asparagus and cook 3-5 minutes, until tender-crisp.
Drain asparagus in a colander reserving 1 cup liquid before draining.
in a saute pan heat olive oil. Add garlic and cook until golden, add asparagus, salt, and pepper and sauté about 1-2 minutes, tossing with oil and garlic.
In a small bowl combine egg yolk, lemon juice, Parmigiano-cheese, 1/4 cup reserved asparagus liquid, salt, and pepper.
After pasta is drained return to pot and mix with egg mixture.
Cook on medium-low about 2 minutes, until sauce thickens and sticks to the pasta.
Toss in asparagus and mix well. Adjust salt and pepper to taste.
Serve with additional grated cheese.
Garnish with chopped parsley and lemon slices if desired.