How to Make It
Bring a large pot of salted water to a boil. Fill a large bowl with ice and water and set aside nearby. Cook the basil in the boiling water until bright green, about 10 seconds. Transfer to the prepared ice bath using a slotted spoon. Let the basil cool, then drain and squeeze dry.
Pulse the pine nuts and garlic in a blender until finely chopped, scraping down the sides as needed. Add the pecorino, Parmesan, basil, and ½ cup water and process until a puree forms, scraping down the sides as needed. With the machine running, slowly add the oil and process until very smooth. Add ¼ teaspoon salt and pulse twice to combine.
Refrigerate in an airtight container for up to 5 days.