Bring a large pot of salted water to a boil. Add the spaghetti and cook according to package directions until al dente. While the pasta cooks, heat a large skillet over medium heat and add the guanciale. Cook until crispy and the fat has rendered. Remove the guanciale from the pan and set aside, leaving the rendered fat in the skillet.

In a medium bowl, whisk together the whole eggs, egg yolks, Pecorino Romano cheese, and freshly ground black pepper until smooth. If using, add the Parmigiano Reggiano as well.

Before draining the pasta, reserve about 1 cup of the starchy pasta water. Drain the pasta and immediately add it to the skillet with the rendered guanciale fat. Toss to coat the pasta.

Remove the skillet from the heat. Slowly pour the egg and cheese mixture over the hot pasta, tossing constantly and vigorously with tongs. The residual heat from the pasta will cook the eggs into a creamy sauce without scrambling them. Add a tablespoon or two of the reserved pasta water as needed to create a smooth, emulsified sauce.

Add the crispy guanciale back to the pasta and toss to combine. Serve immediately, garnished with extra Pecorino Romano cheese and freshly ground black pepper.