Combine the basil, pine nuts, garlic, and oil in a food processor and blend until a paste forms.
Add the Parmesan cheese, Pecorino cheese, salt, and pepper and pulse until smooth.
This is a crucial moment for your pesto, as you need to season with salt and pepper. Keep in mind that Parmigiano and Pecorino are both dry and salty cheeses, you have to taste your sauce a few times in order not to go overboard when you are seasoning it.
Cover the pesto and keep in the refrigerator until you're ready to use it.
Mix the pasta and the pesto, serve garnished with some grated Parmesan (just a touch), fresh ground pepper and a drizzle of olive oil.
Important: Dose your sauce.... some people like the sauce thin, others like it very dense....which basically means less or more sauce that goes in the pasta...The Genovese tradition is to use a small ladle of the pasta water-the one you boiled the pasta in-and use it to dilute the sauce when you mix it in the pasta.