You will need a dutch oven (or a pan with a heavy-bottom), place the oil in it and set it aside. Note: This will not be the same pan you will cook your spaghetti in.
Whisk the egg yolks, whole eggs, Parmigiano-Reggiano and pepper together. Make certain that your eggs are whisked completely.
Sauté the pancetta until cooked and until the fat has rendered out. Set aside to cool slightly. Remove 3 tbsp of the rendered fat and place into a dutch oven or heavy-bottom pan.
In a separate pot, bring 6 qts of water to boil, add salt.
Once the water is boiling add your pasta and cook 2 minutes less than the instructions on the box call say. Just as the pasta is finished cooking, scoop out 1 3/4 cups of the pasta water with a heatproof measuring cup.
Add 1 cup of the pasta liquid to the dutch oven or heavy-bottom pan and bring to a boil over medium heat.
Drain your pasta in a strainer and pour it into the dutch oven or heavy-bottom pan.
Cook the pasta quickly and keep it moving, stirring constantly. You do not want the pasta to stick to the bottom of the pan. Pasta should be al dente and the water should reduce by half. Take the dutch oven or heavy-bottom pot off of the heat.
Go to your egg mixture and whisk in 1/2 cup of hot water into your egg mixture.
Stream this egg mixture into the dutch oven or heavy-bottom pot and stir constantly so the egg mixture gets evenly distributed. The cheese should melt and the egg mixture will thicken, formiing a sauce that adheres to the spaghetti.
If you find you need to thin your sauce, add the remaining pasta water 1 tbsp at a time.
Fold in your pancetta.
Scoop into bowls. We find tongs work well for doing this.
Sprinkle with extra Parmigiano-Reggiano, parsley and salt andpepper to taste.