Bring a large pot of salted water to a boil, add the bucatini and cook until just al dente (about 9 minutes).

Meanwhile, prepare the sauce. Add eggs, egg yolk and cheeses to a large bowl, whisking to combine. Season with several turns of black pepper and pinch of salt. Set aside. Warm olive oil in a large skillet over medium heat, then add the bacon and cook until crispy (about 5 minutes). Add the garlic and cook until fragrant (about 30 seconds). At this point, if the pasta isn't done cooking, take the skillet off the heat. Return the skillet over medium heat 1 minute before the pasta is done.

Drain the pasta well, reserving 1 cup of the starchy cooking water. Add chicken to the skillet with the bacon and garlic, and cook for 1 minutes, tossing to coat in the bacon fat. Then toss in the bucatini, and cook for 2 minutes.

Drizzle a little of the starchy water into the bowl with the eggs, whisking to combine. This will help temper the eggs, and keep them from scrambling. Pour the contents of the skillet into the bowl with the eggs and cheese and using tongs, toss and stir to combine. The pasta will cook the egg, forming a thick sauce. Drizzle more starchy liquid in as you are stirring to thin out the sauce. The bucatini should be evenly coated in the sauce, but you shouldn't see liquid pooling in the bottom of the bowl. Stir in chopped parsley. Season to taste with salt and pepper (this dish can take a lot of pepper).

To serve, divide carbonara between 4 plates and top with extra parsley, Parmesan cheese and black pepper. Enjoy immediately.