One of Italy's most iconic sauces is made from very simple ingredients. The artistry lies in having high quality cured pork (guanciale, pancetta or bacon), Parmigiano Reggiano or Pecorino Romano, and durum-wheat pasta, and in being able to move quickly to finish the sauce once the pasta is cooked. While some people favor adding a little cream to the sauce, purists are against it. The sauce should get its creaminess from the mixture of eggs and cheese. Similarly, some people add a clove of garlic to the pan with the pork (discarding it when the pork is cooked) but it's usually made without it.
Bring a large pan of water to the boil for the pasta.
Place the pork matchsticks in a large frying pan and heat gently over low heat. As the pork begins to render its fat, raise the heat slightly and sauté until the meat starts to become golden and crisp. Keep the pork and its fat hot but don't let it burn.
Add salt and the spaghetti to the boiling water and cook until it is just al dente.
While the pasta is cooking, place the egg yolks in a bowl and whisk in three-quarters of the grated cheese. Whisk in two tablespoonsful of the pasta cooking water. Season with freshly ground black pepper.
Just before the pasta is done, scoop out a mugful of the cooking water and set it aside. Drain the pasta and quickly turn it into the pan with the pork, stirring it over low heat to distribute the pork fat. Immediately pour in the egg and cheese mixture, stirring over low heat to coat the pasta with the sauce. If it seems too dry, add a few spoonfuls of the hot pasta cooking water and cook for 1 minute more. Serve immediately, with the extra cheese passed on the side.