Place the egg yolks, lemon juice, parmesan, mustard, garlic, basil, salt and pepper in the bowl of a food processor fitted with the steel blade and process until smooth. (About 20 seconds.)
Combine the 2 oils in a measuring cup. With the processor running, slowly pour the oils into the bowl through the feed tube and process until emulsified.
Taste for seasoning. (It is a sauce, so it should have ia good amount of seasoning.)
Store the sauce in the refrigerator for at least 3 days. A mason jar works well.