Bring a large pot of salted water to a boil. Add spaghetti and cook 9–10 minutes, just before al dente. Reserve 2 cups pasta water and drain.
Heat a large heavy skillet over medium heat. Add pepper and cook, swirling the pan, about 1 minute until fragrant. Add butter.
Add ½ cup pasta water to the pan and bring to a simmer. Add drained pasta and toss, adding more pasta water as needed.
Remove pan from heat. Add both cheeses, stirring and tossing with tongs until melted (goal is to melt, not cook the cheese). Add a splash more pasta water if the pasta seems dry.
Plate into warm bowls, grind fresh pepper, and sprinkle extra cheese on top. Serve immediately.