Bring a large pot of salted water to a boil. Add pasta and cook until al dente.
Meanwhile, melt the butter in a large frying pan over medium heat. Add the pepper and cook, swirling the pan, until it begins to sizzle, about 1 minute. Set aside.
When the pasta is al dente, use tongs to transfer it to the frying pan and place the pan over low heat. Add Parmigiano and 1/2 cup of the pasta cooking water to the frying pan and, with tongs, immediately toss the pasta until the cheese melts; you want to keep everything moving so the cheese doesn't sink to the bottom of the pan and start clumping. If a film of cheese begins to form on the bottom of the pan, reduce the heat and continue tossing. Add more pasta cooking water as necessary; the sauce should be creamy and emulsified and coat each strand of the pasta.
Remove from the heat, stir in Pecorino, and toss to coat. Transfer to bowls and serve hot.