Recco, which might be called the culinary capital of eastern Liguria, is known for this pasta. Handmade trofie are best, but acceptable commercial trofie are also available.

Steps

1. Make the pesto: Crush the basil, pine nuts, garlic, and a large pinch of salt into a fine paste with a mortar and pestle. Drizzle in the oil, stirring constantly, then stir in the cheese. Put plastic wrap directly on the surface of the pesto, then set aside.
2. (If using a food processor, blend the pine nuts, garlic, and salt into a paste. Add the basil, drizzle in the oil, and process until smooth. Transfer to a bowl and stir in the cheese. Put plastic wrap directly on the surface of the pesto, then set aside.)
3. Make the trofie: Sift the flour and salt together into a mound on a clean surface. Use your hand to make a well in the center, then pour in about 1¼ cups water. Flour your hands, then knead the flour and water together with both hands until the dough is soft and no longer sticky. Push the dough to one side, clean the surface and your hands, then flour both again and knead dough again, adding more flour if necessary, until it is very smooth, about 2 to 3 minutes. Cover the dough with a kitchen towel and set aside for half an hour.
4. Clean and flour hands and surface again. Pinch off a pea-size piece of dough and roll it away from you with the palm of your hand to form a fat matchstick. Turn your hand up at a 45-degree angle, then gently roll the matchstick back toward you to form a spiral shape with two pointed ends. Repeat to use all the dough. (This may take practice; expect to throw away some ill-formed trofie at first.) As trofie are made, transfer them to a lightly floured, parchment-lined baking sheet.
5. Cook the vegetables: Cook the beans in a pot of boiling salted water over high heat until tender, about 5 minutes. Remove the beans with a slotted spoon, then add the potatoes to the same water and boil until soft, about 15 minutes. Remove the potatoes with a slotted spoon, allow to cool, and then slice thinly.
6. Cook the pasta: Cook the trofie in the same water over high heat until they float to the surface, about 4 minutes. Dilute pesto with 2 Tbsp. pasta water, then drain the pasta, transfer it to a large bowl, and mix with the pesto, potatoes, and beans.