Bring 2 1/2 quarts water (about 10 cups) to a boil in a large deep skillet. Add 1 tablespoon salt and the pasta and cook until just al dente (1 to 2 minutes less than the label directs).
Meanwhile, put the peppercorns in a small resealable plastic bag and coarsely crush with the flat side of a meat mallet or the bottom of a skillet. Place the crushed peppercorns in a fine-mesh sieve and discard any dust that falls through. Heat the olive oil in a small skillet over medium heat. Add the crushed peppercorns and cook until sizzling, 1 to 2 minutes. Let cool slightly, then transfer to a large bowl.
Using tongs, remove the pasta from the boiling water and transfer to the bowl with the peppercorn oil along with 2 tablespoons of the cooking water. (Reserve the remaining cooking water in the skillet.) Toss the pasta until coated. Let cool slightly, 1 to 2 minutes. (If the pasta is too hot, the cheese will clump and not melt.)
Add the cheese to the bowl with the pasta in three batches, tossing well and adding a few tablespoons of the reserved cooking water with each addition; make sure most of the cheese is melted before adding more. Continue adding the cooking water, a few tablespoons at a time, until the pasta is coated in a thick creamy sauce.
Photograph by Ryan Liebe