Bring a large saucepan of salt water to a boil (about 1 teaspoon of salt) over medium-high heat. Once the water has come to a boil, add the pasta and cook according to the package directions until it is al dente. Drain pasta, reserving one cup of pasta water.
Prepare the egg and cheese mixture and cook the guanciale while the pasta is cooking.
In a medium mixing bowl, whisk together the egg, egg yolk, grated Parmigiano Reggiano, and black pepper until a creamy sauce has formed. Set it aside.
Cook the cubed guanciale in a large skillet over medium heat, stirring occasionally to ensure even cooking. Cook until crispy, and the fat has been rendered, then turn off the heat.
Once the pasta has been cooked and drained, reheat the guanciale over medium-high heat, then add the drained pasta to the pan. Toss the guanciale and pasta in the rendered fat to coat the pasta, then turn off the heat.
With the heat off, add the egg and cheese mixture and stir. The texture should be creamy, but you can add the reserved pasta water a tablespoon at a time if needed for creaminess.
Serve immediately with additional grated cheese, pepper, and any remaining guanciale in the skillet.