Bring 2 quarts of water to a boil in a large Dutch oven or pasta pot. Season with 1/2 teaspoon kosher salt. Add the spaghetti and cook until just shy of al dente. Before transferring the pasta, reserve 1 cup of the pasta cooking water.

While the pasta cooks, toast the peppercorns in a dry skillet over medium heat until fragrant, about 2 to 3 minutes. Transfer to a mortar and pestle or spice grinder and coarsely grind. Reserve a little pepper for serving, if desired.

Place the Pecorino Romano in a food processor and process until finely ground, about 2 minutes. With the processor running, slowly add 1/4 cup hot tap water until a thick, smooth paste forms, similar to Greek yogurt. If needed, add additional hot water 1 tablespoon at a time until smooth.

Transfer the cheese paste to a large heatproof serving bowl. Stir in the toasted pepper and 2 tablespoons of the reserved pasta water.

Using tongs, transfer the hot pasta directly to the bowl. Add 1/4 cup of the reserved pasta water and half of the cheese paste. Toss vigorously until the cheese begins to melt and coat the pasta.

Add the remaining cheese paste and continue tossing, adding more reserved pasta water a few tablespoons at a time, until the sauce is smooth, glossy, and evenly coats the pasta. The sauce should cling to the noodles without pooling in the bottom of the bowl.

Divide among warmed serving bowls and serve immediately with additional Pecorino Romano and freshly cracked black pepper, if desired.