Heat olive oil in a largeskillet over medium heat; add guanciale or bacon. Cook, turning occasionally, until browned and just slightly crispy (5 to 10 minutes). Remove from heat and drain off some excess fat if you wish.
Bring a large pot of salted water to a boil. Add spaghetti and boil, stirring occasionally until al dente (about 9 minutes). Drain and return to pot.
Whisk eggs, half of the cheese, and pepper in a bowl until smooth and creamy. (If you prefer to cook your beaten eggs before pouring over the pasta, gently heat over LOW heat, stirring constantly so the egg mixture does not get too hot.)
Pour egg mixture over pasta, stirring quickly, until creamy and slightly cooled.
Stir in guanciale/bacon and top with remaining cheese and more pepper to taste.