Cook the potatoes
Add the potatoes to a saucepan with water to cover. Bring to a boil over high heat, reduce the heat to medium, and cook until tender, about 20 minutes. Drain and let cool. Peel the potatoes and cut into small cubes.
Cook the pasta and green beans
Meanwhile, bring a large pot of water to a boil. Add 2 tablespoons salt and the pasta. Cook, stirring occasionally to prevent sticking, until al dente, according to the package directions. Add the beans about 5 minutes before the pasta is ready.
Finish the dish
Drain the pasta and beans, reserving about ½ cup (4 fl oz/125 ml) of the cooking water. In a large serving bowl, mix a few tablespoons of the cooking water with the pesto. Add the potato cubes, drained pasta, and beans and toss to combine. Add as much of the remaining cooking water as needed to achieve a nice coating consistency for the sauce. Serve, passing the cheese at the table.