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Put the avocados, lemon zest and juice, garlic, parsley, half the basil, 4 tablespoons of water and some seasoning in a food processor. Blitz until smooth, taste and adjust the seasoning, if necessary. Put in a bowl and chill if you are not using it straightaway.
Cook the penne following the pack instructions. Drain and toss with the avocado sauce, adding a little of the pasta water for a creamier sauce. Sprinkle over the cheese, stir. Serve on a platter or in bowls, scattered with the remaining basil, and extra cheese.