Add water and salt to a large pot and bring to a boil.
Meanwhile, place peppercorns in a small saucepan over medium heat. Swirl peppercorns around in the pan until lightly toasted and fragrant, about 1 minute.
Using a mortar and pestle, crush the peppercorns. Set aside.
Add bucatini to the boiling water and cook until halfway done, approximately 6 minutes.
Reserve the pasta cooking water before draining the bucatini. Set both aside.
In a large saucepan, melt the butter.
Add 1 tablespoon crushed black pepper and stir.
Add a large ladle of pasta water to the pan and bring to a simmer.
Add the bucatini and move it around in the pan using tongs. Cook until al dente, adding more pasta water if necessary, about 3 minutes.
Add pecorino and stir until fully melted and incorporated into the sauce.
Serve immediately with more pecorino and pepper, if desired.