Place dry spaghetti in an 11-inch straight-sided sauté pan and add enough cold water to just cover the pasta. Add salt, cover, and set over medium-high heat.
When the water reaches a boil, remove the lid and decrease heat to medium to maintain a simmer. Stir the pasta every minute or so, making sure the ends of the noodles stay submerged and that they aren’t sticking together.
While the pasta cooks, begin the sauce. Combine 2 tablespoons of the black pepper, 3 1/2 ounces (100 grams) of the pecorino, all of the Parmesan, and the olive oil in a large, deep mixing bowl. Work ingredients together in the back of a wooden spoon until a thick paste forms.
When the pasta has been simmering for 5 minutes, ladle 1 cup of the cooking water into a measuring cup, then slowly drizzle 3/4 cup of the water into the cheese paste, mixing until smooth.
After the pasta has been cooking for 10 minutes, start checking for doneness; you want it to be just barely al dente as there will be some carryover cooking in the next step.
Use tongs to lift the pasta out of the cooking water and allow to briefly drain before adding to the cheese paste. Grasp the pasta with the tongs and stir vigorously for 2 minutes—this is not a light toss. The pasta will continue to release starch and the sauce will emulsify. Around 90 seconds into tossing, you will see a cheesy water magically transform into a proper sauce. If you find your sauce is clumping, add a bit more of the pasta water and continue tossing.
Portion pasta into bowls, top with reserved pecorino, and serve immediately.