Put the basil leaves, garlic cloves, pine nuts, and Parmesan cheese into a food processor. Then pulse the processor for a few seconds to coarsely chop all the ingredients.
Add the Pecorino cheese. Then pulse again to incorporate it into the pesto until the mixture is finely minced.
With the processor still running add the olive oil in a very slow but steady stream. The pesto will become smooth and creamy. This process will take a few minutes.
Taste the finished pesto one last time and add salt if needed.
Instructions for Mortar and Pestle
Pound the garlic cloves together with just a pinch of salt with the mortar and pestle until it’s a chunky paste.
Add the basil leaves in 3 batches pounding them together with the garlic with each addition until a paste forms.
Add the pine nuts and grind them into the paste until well combined.
Add the cheese a third at a time pounding and grinding it into the pesto before adding the next third.
Drizzle 1 tablespoon of the olive oil at a time into the mortar and then pound it into the sauce with the pestle. For ½ cup of oil that means that you will do this 8 times. Once all the oil has been added the pesto should be a creamy emulsified mixture.