Great if you can get fresh basil - tear with your hands to minimize 'blackening' the basil.
3 cups packed tender young basil leaves (Genoa or Thai has a slight 'anise' flavor)
3 heaped T pine nuts (you can also use walnuts)
1 1/2 t coarse salt
1/2 cup extra-virgin olive oil or more to taste
3 fat garlic cloves, crushed with the flat blade of a knife and very finely minced
1/2 cup freshly grated cheese, preferably a mixture of Parmigiano and an aged Tuscan pecorino or pecorino sardo; otherwise, use all Parmigiano.
3 medium potatoes, peeled and sliced not more than 1/4 inch thick
1 lb trenette, linguine, tagliatelle, or other long flat thin pasta
PUT the basil, pine nuts, and salt in a food processor or blender and process steadily while you add the oil in a thin but constant stream. The sauce should achieve the consistency of a slightly grainy paste but not a fine puree.
ADD the garlic and process very briefly, just to mix.
WHEN the sauce is the right consistency, transfer it to a bowl and, using a spatula, fold in the grated cheese. (If you're using a mortar, just continue to work in the cheese with the pestle.)
IF the sauce is too thick, work in more olive oil. Taste and adjust the seasoning.
PESTO can be made ahead and stored in the refrigerator for a few days or in the freezer for a few weeks. If you plan to store it, leave the cheese out. Transfer the pesto to a refrigerator container, pour a thin film of oil over the top, cover, and store. When you're ready to use it, let the pesto thaw if necessary, then stir in the oil on top and the grated cheese.
BRING the lightly salted water (use lots of water for cooking pasta) to a rolling boil. Drop in the potato slices and boil for 5 minutes, then drop in the pasta. Stir with a wooden spoon, cover the pot, and bring back to a boil. Uncover and boil the pasta and potatoes together for 10 to 12 minutes or until the potatoes are thoroughly cooked and the pasta is done to taste.
WHILE the pasta is cooking, stir a couple of tablespoons of hot pasta water into the pesto and rinse a big pasta bowl with boiling water to warm it.
AS soon as the pasta is done, drain it, turn it into the warmed bowl, and pour the pesto over it. Toss gently to mix the pesto and serve immediately, passing more cheese if desired.