Bring the water to a boil in a large pot. Add the kosher salt, then add the spaghetti. Cook, stirring occasionally, until the pasta is just shy of al dente, following the package timing as a guide.

Before draining, scoop out 1 1/2 cups (360 mL) of the starchy pasta water. Drain the spaghetti; do not rinse it.

While the pasta cooks, warm a large skillet over medium heat. Add the black pepper and toast for 30 to 60 seconds, stirring, just until fragrant. Add 1/2 cup (120 mL) of the reserved pasta water and let it gently simmer for about 30 seconds. Turn off the heat.

In a medium bowl, whisk the finely grated Pecorino Romano with 1/3 cup (80 mL) warm reserved pasta water until it becomes a thick, smooth paste. Do not use actively boiling water, which can make the cheese seize.

Add the drained spaghetti to the skillet with the pepper water. Toss well for 30 seconds so the pasta absorbs some liquid.

With the skillet off the heat, add the cheese mixture and toss continuously with tongs. Add the remaining reserved pasta water a splash at a time, tossing after each addition, until the sauce is glossy, creamy, and coats the spaghetti. If the skillet becomes too hot, wait 1 minute before adding more cheese sauce.

Taste and add a little more black pepper if desired. Serve immediately, topped with extra finely grated Pecorino Romano and black pepper.