Prepare the pesto according to this recipe, using ⅓ cup olive oil.
Bring a large pot of salted water to a boil. Prepare the spaghetti according to the package instructions, cooking until al dente. Reserve 1 cup of the pasta cooking water, then drain.
Transfer the pasta to a large skillet. Add the pesto and ½ cup*** of the starchy pasta water and toss to coat, adding more pasta water as needed to coat the pasta in a lightly creamy sauce. If desired, gently warm the pasta over low heat.
Season to taste and serve topped with freshly grated Parmesan and lots of basil.