Slice the guanciale into strips about 0.5 cm thick. Set aside while preparing the other ingredients.
In a large mixing bowl, combine the whole eggs and additional yolks. Whisk until fully blended, then incorporate the finely shredded cheese and ground black pepper. Set this mixture aside.
In a large pot, bring approximately 4 liters (4 quarts) of water to a rolling boil. Season generously with the coarse salt. Add the spaghetti and cook according to the package instructions until al dente.
Before draining the pasta, reserve approximately 1 cup of the starchy cooking water. Drain the pasta promptly.
While the pasta cooks, heat a non-stick skillet over medium-high heat. Add the guanciale and cook for 4–5 minutes until golden and crisp, allowing the rendered fat to accumulate in the pan. If using garlic, stir it in during the final minute of cooking.
Transfer the hot, drained pasta directly into the skillet with the guanciale. Toss well to ensure the pasta is fully coated with the rendered fat.
Quickly move the pasta and all pan contents into the bowl with the egg-cheese mixture. Immediately pour in 1/2 cup of the reserved pasta water. Using the handle of a wooden spoon, stir the pasta vigorously for about a minute until the sauce thickens into a creamy consistency that envelops the noodles.
Serve the pasta promptly in warmed serving bowls. If desired, sprinkle with additional grated cheese, a dash of pepper, and a small amount of chopped parsley.