Wash the basil leaves and dry them very well by placing them on paper towels.
Crush the garlic and place it in the food processor along with a pinch coarse salt, half of the basil and 1 tbsp of olive oil. Blend at a low speed and pulse rather than continuously mixing at high speed.
Add the rest of basil and continue pulsing for a few seconds.
Then add the pine nuts, Parmesan, pecorino and 3 tbsp olive oil. Mix again intermittently, until everything is well blended and somewhat smooth with a bit of graininess.
Empty the pesto into a bowl and dilute it by adding 1 tbsp olive oil, little by little until you have a soft texture.
To avoid overheating you can also place the food processor bowl and blades in the refrigerator for an hour before.
Weigh your basil leaves. Providing weight rather than cups is the most precise way to measure the amount of basil you need. Make sure you remove the stem, only use the leaves.
Place the pesto on an airtight jar covering the sauce with a layer of olive oil and refrigerate for up to 3 days.
You can freeze pesto by placing single portions (about ¼ cup) on a sheet of plastic wrap. Wrap and then wrap again with another sheet of plastic wrap. Store in the freezer for up to 3 months. Thaw at room temperature.