This is a new and improved method taken from a cooking class I took when visiting Liguria in 2023.
Crush the garlic with a pinch of salt to help break it up
Add the pine nuts and continue to crush but not completely to a paste
Add the leaves slowly in batches, crushing them. Once they're all integrated, use a circular motion around the edge of the pestle to form a smooth paste.
Once you've reached the desired consistency, add the two cheeses and mix, then slowly incorporate olive oil until you reach the desired pesto consistency.
Since Pesto is heavy in oil, it can have the tendency to split on you and leave an oily mess on the bottom of your plate or bowl if you try to stir it into pasta that's in a hot sauce pan, like you might do with many other pasta sauces.
In order to combat this, its best to cook your pasta (preferably trofie pasta if you can find or make that shape) and drain it then reserve it off to the side. In a bowl, mix your pesto with a bit of hot, but not boiling water. I use somewhere around 140 to 150 degrees Fahrenheit. If you have an electric kettle with adjustable temperature, it's ideal for this purpose. Mix your pesto with water in your bowl, until you achieve something thats loose but not overly watery (you can always add more water but you can never take it away), then add the cooked pasta and stir to coat the pasta. This should result in a sauce that adheres nicely to the pasta without splitting.