Wash the basil leaves very gently with a damp cloth or kitchen paper to avoid oxidizing the leaves.
If you wish to make pasta with the pesto, you can start boiling the water when you begin preparing the sauce. If you also want to add potato and green beans, dice the potato and beans into small cubes. Place the potatoes in a small pot with cold water, salt the water to taste when it starts boiling, and when the potatoes are about a minute away from being fully cooked, add the green beans so they remain al dente.
Place the garlic, from which we have previously removed the inner sprout, and a pinch of coarse salt in a mortar and crush the garlic until it reaches the texture of a cream.
Add the basil leaves and another pinch of salt and crush the leaves against the walls of the mortar with the pestle until they have the consistency of an intense green purée.
Incorporate the pine nuts until they form a homogeneous paste with the rest of the ingredients in the mortar.
Add the grated cheeses without stopping crushing and stirring the pesto so that everything mixes well.
Incorporate the oil and bind it well with the rest of the ingredients.
Add a ladle of cooking water to the pesto to make it a bit less dense, stir, and pour the pasta into the pesto.
Mix the pasta well with the pesto, and it's ready to serve.