My recipe (speaking as someone from Rome, my tastes might be different):
Ingredients:
- 1 egg yolk per person + 1 full egg. (E.g., 2 people = 2 yolks + 1 egg)
- pecorino romano (a lot). I put also a 20%-ish of parmigiano to balance the taste.
- black pepper, freshly ground.
- guanciale (traditionally, I live abroad and often use pancetta - which is less fat and might require a little bit of olive oil)
Usually you want spaghetti or maybe rigatoni, fettuccine or similar (like OP) tend to suck too much the sauce and are also heavier (it makes sense that they used many full eggs).
Preparation:
You beat the eggs and add scraped pecorino until the result is thick.
You add pepper and a bit of salt to it as well and mix.
In a pan with no oil or butter you put the guanciale and you let it sweat. You let it fry in its own fat until it's like you want it.
You can take a couple of teaspoons of fat and add it to the egg and pecorino mix.
Depending on your taste, you can remove a bit of fat.
You put water boiling and you salt it generously.
You boil pasta, and take it out approximately 2 minutes before the official cooking time. You add the pasta in the pan with the guanciale, and you add cooking water into it to continue the cooking while you mix (few water, multiple times, bit by bit).
With the pasta still wet, you add it to the container where the egg mix is (not on fire). Better too dry (in which case you add a bit of cooking water) than too liquid (cannot be repaired easily, you will have to drop it in the pan and let it dry).
You mix vigorously and you should have the egg sauce perfectly attached to the pasta.
If you put enought pecorino in the sauce, you probably won't need additional one on top.
That's it.
There are people who do it very differently, for example there are those who mix egg with so much pecorino that they make a solid ball that they add to the pan while finishing the cooking of the pasta and they melt it with cooking water.
Either way, carbonara (and cacio e Pepe) are extremely simple recipes that have a tricky process easy to mess up, and it takes a few attempts to get it as you want it.