Place the bowl and blades of a food processor in the refrigerator or freezer until the tools are very cold, about 15 minutes.
Meanwhile, get the basil leaves ready by washing them in cold water.
Place the clean basil in a large bowl with plenty of ice for 3-4 minutes.
Remove leaves from the ice and dry them very well in a kitchen towel. Important: The basil leaves must be very dry.
Remove bowl and blades from the refrigerator or freezer and place basil leaves, garlic, pine nuts, and grated Parmigiano in the food processor bowl.
Pulse a few seconds in the food processor.
Add salt and Pecorino cheese to the bowl.
Blend all ingredients in the food processor for about 1 minute.
Add olive oil to the bowl and blend for about 5 minutes at medium speed, and at intervals: blend a few seconds, stop and start again until you see a creamy green pesto sauce. Work quickly as you do not want the pesto to heat up.
Serve Pesto Genovese over pasta (you may want to add a tablespoon or so of the pasta water to the Pesto to thin it out a bit, as needed) or as a spread on toasted bread as an appetizer. Yield: About 1 cup; 6 servings. 33 cal per serving.
Store Pesto Genovese in the refrigerator, in an airtight container for 2-3 days, taking care to cover the sauce with a layer of extra virgin olive oil.
It's possible to freezer pesto in small jars, again covered with a thin layer of olive oil, and then defrost it in the refrigerator or at room temperature.