Cut off the tips of the green beans. Peel the potatoes and cut them into about 1 cm cubes.
Prepare the pasta according to the instructions on the package. Add the potatoes to the boiling water along with the pasta, and add the green beans a few minutes before the end of cooking.
Meanwhile, prepare the pesto. To make pasta with pesto sauce, basil leaves should not be washed, but gently wiped with a soft cloth; it is better to choose narrow-leaved basil.
Place the peeled garlic in a mortar with a few grains of coarse salt. Start mashing, and when the garlic turns into a cream, add the basil leaves.
Press the fresh basil against the sides of the mortar by turning the pestle from left to right, while simultaneously turning the Mortar in the opposite direction (from right to left).
Continue this process until a bright green liquid comes out of the basil leaves, then add the pine nuts and start mashing again to turn them into a cream.
Add the cheeses little by little, stirring, to make the sauce even creamier, and finally add the extra virgin olive oil, which will be poured, always stirring with a pestle. Mix the ingredients well until the sauce is smooth.
When the pasta is almost cooked, set aside a small cup of water for cooking. Drain the al dente pasta along with the potatoes and green beans.
Place the pesto in a large skillet and pour over the pasta, stirring while cold and diluting with a little cooking water to make a creamy sauce.
Your Genoese pesto pasta with potatoes and green beans is ready. Serve it with grated cheese, if desired (Parmesan or Pecorino).