Bring a large pot of water to a boil and add salt. Add the bucatini and cook, stirring occasionally so the pasta doesn't stick together.
While the pasta is cooking, heat a 12-inch skillet over medium high heat and add the 1 tablespoon olive oil. Add the guanciale and cook, stirring occasionally, until it begins to crisp, 2 to 3 minutes. When the guanciale is almost completely crispy, carefully add 1/4 cup of the pasta water and scrape all of the tasty bits from the bottom of the skillet with a wooden spoon. Remove the skillet from the heat.
Whisk together the egg whites in a medium bowl until they start to get frothy. Add lots of pepper, the parsley, extra-virgin olive oil and Parmesan. Pour over the guanciale in the skillet, stirring vigorously to incorporate but not scrambling the eggs.
When the bucatini is al dente, remove it from the water and toss with the sauce until well coated.
Divide the pasta into 2 bowls, top each one with an egg yolk and serve immediately, letting your family or guest mix in the yolk at the table for that extra decadence!