A classic Roman dish, alla carbonara — in the style of the coal worker — has several origin stories, notably turning the copious amounts of ground black pepper to metaphorical flakes of coal. We think of it as the best way to have your bacon and eggs. Here, guanciale, the gift of the pork jowl, ably fills the role of bacon.
Bring a large pot of water to boil over high heat.
Place the guanciale in a medium skillet with the olive oil and garlic cloves. Set over medium heat and cook, stirring frequently, until fat has rendered from the meat and edges are crispy, 5 to 6 minutes. Deglaze the pan with wine, stirring up any browned bits and letting the wine boil away. Turn off the heat. Discard garlic cloves.
Add a generous amount of salt to the water when it boils, then drop in pasta and stir once.
In a large serving bowl, beat the eggs and yolks lightly with a fork. Mix in both cheeses and the black pepper.
Drain the pasta when it’s tender but firm (al dente), reserving 1/2 cup pasta water. Put the pasta in a large serving bowl.
Slowly drizzle some of the reserved pasta water into the eggs while whisking to make a silky, smooth sauce.
Pour the sauce over the hot, steamy pasta and toss until nicely coated. Add more of the reserved pasta water if you need to thin the sauce.
Quickly reheat the guanciale over high heat, about 30 seconds, then pour meat and drippings over the pasta. Toss and serve.