Cut the pasta dough into pappardelle.
In a large pot of boiling, salted water, cook the pappardelle, stirring often, just until al dente, about 3 minutes.
Meanwhile, in a large, heavy skillet over medium heat, melt the butter. Stir in the pesto. Using a spider or small strainer, transfer the pasta from the pot of water to the skillet. Add 1/2 cup of the pasta cooking water; toss gently. Sprinkle with the cheese; season with salt and pepper. Toss again, adding more cooking water if necessary to thin the sauce.