Briefly soak and wash the basil in cold water, and gently pat it thoroughly DRY with paper towels.
Put the basil, olive oil, pine nuts, chopped garlic, and an ample pinch of salt in the processor bowl, and process to a uniform, creamy consistency.
Transfer to a bowl, and mix in the two grated cheeses by hand. It is worth the slight effort to do it by hand to obtain the notably superior texture it produces.
When the cheese has been evenly amalgamated with the other ingredients, mix in the softened butter, distributing it uniformly into the sauce.
When spooning the pesto over pasta, dilute it slightly with a tablespoon or two of the hot water in which the pasta was cooked.
Freezing pesto: Make the sauce through to the end of Step 2, cover with a layer of olive oil and freeze it without cheese and butter in it. Add the cheese and butter when it is thawed, just before serving.