Lightly crush the garlic to loosen the skin, then peel and discard the skin.
Briefly soak and wash basil leaves in cold water, then pat them gently but thoroughly dry.
Put basil, garlic, pine nuts, and coarse sea salt in a mortar. Grind with a rotary motion against the side of the mortar until a paste forms. Add both grated cheeses and grind them evenly into the mixture.
Add olive oil in a very thin stream, beating it in with a wooden spoon. When incorporated, beat in the softened butter until evenly blended.
When spooning over pasta, dilute the pesto slightly with 1 to 2 tablespoons of the hot pasta water.