Cook the pasta in a large pot of salted boiling water until al dente. Reserve about 1 cup of pasta water.
Place the basil, pine nuts, garlic, salt, and cheeses in a food processor. Pulse in intervals for about 1-2 minutes, pausing and scraping down the sides as needed.
While pulsing, add the extra virgin olive oil and continue to pulse until smooth.
When the pasta is ready, drain it and mix in the pesto, adding a bit of pasta water to adjust the consistency of the sauce.
If you would like to make your pesto with a mortar and pestle, use the same ingredients as above and proceed as follows:
Begin by placing the garlic cloves and pinch of salt in the mortar. Grind to a smooth paste. Add the pine nuts and continue to grind until the consistency is uniform.
Add the basil leaves (working in batches if necessary) and gently push the pestle on the basil leaves in a twisting motion to release the flavor.
Once the basil has broken down, add the cheeses and stir until incorporated. Drizzle in the extra virgin olive oil and finish off with the pestle until the sauce is consistent throughout.