Place the potatoes in a medium saucepan, add salted water to cover generously, and bring to a boil. Boil gently until the potatoes are tender; drain. Cut the potatoes in half, and set aside.
Meanwhile, cook the beans in a large saucepan of boiling salted wter until tender, 4 minutes. Have an ice bath ready. Drain the beans, plunge into the ice bath just to cool, and drain again. Set aside.
Bring 6 quarts of water to a boil in a large pot, and add 2 tablespoons salt. Cook the pasta until al dente; drain.
Pour the pasta into a warmed bowl, add the beans, potatoes, and pesto, and toss to coat the pasta and to warm the beans and potatoes; do not return to the heat. Serve with grated Parmigiano on the side.
Combine the pine nuts, basil, garlic, and salt in a large stone mortar and grind with the pestle until the mixture forms a paste. Slowly drizzle in the olive oil, beating all the while with a wooden spoon. Add the Parmigiano 1 tablespoon at a time, beating until the mixture forms a thick paste. The pesto can also be made in a food processor. The pesto can be stored in a jar, topped with a thin layer of extra-virgin olive oil, for several weeks in the refrigerator.