There are four "Roman classic" pasta dishes that you can find at most restaurants/trattorias in Rome:
Cacio e pepe - (Pecorino Romano) cheese and pepper, like if maccaroni and cheese died and went to heaven
Gricia - add guanciale (cured pork jowel), which is like if bacon died and went to heaven
Carbonara - add egg yolks for an unctuous, creamy sauce punctuated by the crackle of the guanciale
Amatriciana - remove the yolks and add tomato sauce; a satisfying, cheesy tomato sauce that is the easiest of the four to make
Even though the "recipes" are simple (and rarely written down in Italy), it's important to get high-quality ingredients and the execution is tricky: the first few times I made cacio e pepe it was a lumpy, sad mess.
Here's what has worked for me so far:
2 main "tricks":
Grate the cheese with a "star" box grater - NOT a Microplane, which created the lumpy, sad mess mentioned earlier. A box grater can create the powdery, fine cheese that you need to get a good emulsion (I use this Oxo grater; Serious Eats did a deep dive into graters and cacio e pepe)
Add cold water to the cheese before stirring - ideally ice cold water but even room-temperature is OK; basically you make a "paste" with the cheese, so it's less likely to get lumpy or stick together when the hot pasta/pasta water hits it (I learned this from Flavio de Maio of the famed Roman restaurant Flavio Velavevodetto, by way of Elizabeth Minchilli, who also has a slower video with her explaining things)
(there's also a "foolproof" or "restaurant" version by Luciano Monosilio which involves corn starch and a blender; Ethan Chlebowski goes into more detail about the science behind this version)
Traditionally this is made with tonnarelli, which is like spaghetti but with a square instead of circle cross-section...that being said, spaghetti is fine (do make sure it's a good quality one though)
Ingredients
~2 cups grated Pecorino Romano (you can also do 1/2 pecorino, 1/2 parmigiano reggiano)
1 pound pasta
~2 tsp fresh-ground black pepper (more or less to taste)
Instructions
Cook the pasta (doesn't need much salt since the cheese has a lot of salt...also use minimum water in the pot to raise the starch content in the water, which helps your emulsion)
Grate the cheese; add pepper to the cheese and stir to combine
Add a small amount of cold water to the cheese, and stir vigorously to create a paste
When the pasta is al dente, reserve some of the pasta water and drain
Place the pasta in a large bowl, add the cheese paste, and some of the reserved pasta water; quickly stir (I've found that a large set of tweezers works less, but you can use chopsticks or a meat fork) to create an emulsion
Serve immediately with extra cheese & pepper on top
This recipe is based on the one made by Luciano Monosilio, who is known as the "king of the carbonara" in Italy; we were lucky enough to try his version at Luciano Cucina Italiana (now one of our favorite restaurants in Rome), and it's amazing!
Do NOT try to do more than 1 pound of pasta; every time I've tried, it came out as a clumpy mess
You can use bacon instead of guanciale, but it won't be quite the same...
You can watch Chef Monosilio cook this recipe in this video...and Alex "French Guy Cooking" Aïnouz goes through the technique in more detail in this video
Ingredients
1 package spaghetti (or other pasta; rigatoni are often used traditionally)
50 grams (~1/2 cup) grated cheese, on box grater star side (I often use all pecorino romano, but original recipe calls for 20g pecorino & 30g grana)
~5-6 oz guanciale, with “skin” removed and cut into ~1 cm cubes
in San Francisco, we've found some amazing guanciale from Alle Pia Salumi at Limoncello
4 egg yolks
Instructions
Brown guanciale on all sides in a pan on ~medium heat, turning down to low as necessary so it doesn’t burn (and stays soft on the inside); meanwhile, start water boiling for pasta
Drain rendered guanciale fat into a bowl & set guanciale aside to cool to room temperature
Cook pasta (as with cacio e pepe, above, relatively little salt & water helps) until al dente; reserve some pasta water
Whisk egg yolks with cheese, then whisk in a small amount of hot pasta water to loosen & help “temper” the yolks
Whisk egg mixture over a bain marie, slowly adding in half of the guanciale fat to create an emulsion
Once egg mixture is sufficiently thick (like a custard), remove from bain marie and whisk in some more pasta water (to further “temper” the yolks & make it easier to coat)
Toss the sauce with the pasta (mixing with a set of tweezers works well)
Add guanciale on top & season with pepper (and more cheese) to taste