In a large pot of salted boiling water, boil pasta, potato, and green beans until pasta is al dente and potato and green beans are very tender. Drain, reserving 1 cup cooking water, and transfer pasta, potato, and green beans to a large mixing or serving bowl.
Add pesto sauce to pasta along with ¼ cup pasta cooking water. Toss well to emulsify pesto and pasta water into a creamy sauce. Add more pasta water, 1 tablespoon at a time, as needed, if pasta is too dry. Drizzle in fresh olive oil, if desired. Serve with Parmigiano Reggiano on the side.
