Add the pasta to the pot and cook until al dente, or according to the package instructions. Do not drain the water after cooking — you need the cooking water for the sauce.
Meanwhile, in a blender, place the avocado halves, olive oil, lemon juice and Parmesan cheese.
Pulse briefly to combine, and then stream in 1/2 - 3/4 cup of the pasta cooking water until a smooth, creamy sauce forms. Taste and add salt and pepper, if needed.
Combine the pasta and avocado sauce and serve with extra Parmesan on top.