Place the garlic and coarse sea salt in a large mortar and pound to a smooth paste, 1–2 minutes.
Add the pine nuts and pound until creamy and fully integrated, 1–2 minutes.
Add the basil leaves in 3–4 additions, pounding with a gentle, rotating motion until a vivid green, slightly coarse paste forms, 3–5 minutes total.
Add the Parmigiano Reggiano cheese and Pecorino Fiore Sardo cheese; pound to combine into a thick paste, about 1 minute.
Gradually drizzle in the extra-virgin olive oil while working the pestle in a circular motion to emulsify. Continue until glossy and spoonable, 1–2 minutes. Use immediately, or refrigerate in an airtight container for up to 3 days.