If possible, chill a marble mortar and wooden pestle for 10 minutes. Finely grate the cheeses and set aside.
In the mortar, pound the garlic with the coarse sea salt to a smooth paste, 2β3 minutes.
Add the pine nuts and grind until creamy with no large pieces remaining, 1β2 minutes.
Add the basil leaves (washed and thoroughly dried) in 2β3 additions, crushing with a circular motion until a thick, bright-green paste forms, 3β5 minutes.
Add the Parmigiano Reggiano cheese and the Pecorino Sardo cheese; work them in until fully incorporated, 1β2 minutes.
Drizzle in the extra-virgin olive oil a little at a time, stirring until the sauce is glossy and spoonable; it should hold its shape on a spoon without oil pooling.
Taste and adjust if needed. Use right away, or transfer to an airtight container, pressing plastic wrap directly against the surface to limit oxidation. Refrigerate up to 2 days; return to room temperature before using.