Bring a pot of water to boil, season it liberally with sea salt. Fill a large bowl with ice water.
Add the basil to the boiling water for 30 seconds or so and transfer to the ice bath. Drain and dry the basil in a salad spinner or on clean kitchen towels.
Place the EVOO in the food processor and add the basil, Parmigiano-Reggiano, Pecorino-Romano, pine nuts, lemon juice, garlic and some salt and pepper. Pulse process into a fine sauce. Store in an airtight container. Freeze or refrigerate to store for a make-ahead meal. Bring to room temperature to prepare.
To serve, bring a large pot of water to a boil, salt the water and cook the pasta to al dente. Reserve a cup of the salty cooking water just before draining the pasta.
Place the pesto in serving bowl. Add the starchy water and stir to combine, then add the pasta and toss 1 to 2 minutes to coat evenly. Adjust the seasoning and serve.