Bring a large pot of salted water to boil. Meanwhile, wash and pat dry the basil leaves and gently tear them into large pieces. Place the butter and oil in a large skillet over medium heat. When the butter is melted, add the basil and some salt and pepper.
Cook the pasta until al dente (a little less than the package instructions). Drain the pasta, but retain some of the cooking water. Add the pasta to the skillet with the butter and basil and mix together over medium heat for a few seconds. If it seems dry, add some of the cooking water. Serve immediately with a sprinkle of Parmigiano.
Because different brands of pasta cook up to different volumes, you may need to add more butter or olive oil to the pasta once it is dressed.