Fill a large pot with 3 quarts of water and 1 tablespoon of salt and place it over high heat. Bring the water to a boil. Add the pasta to the boiling water and cook following the package directions for al dente.

Meanwhile, put the bacon in a large sauté pan or deep skillet over medium heat and cook until crisp. Drain the bacon on paper towels. Set the pan with the bacon drippings aside.

In another bowl, combine the egg and egg yolks with 1/3 cup of Parmesan cheese; whisk to blend. Add the drained bacon and set aside.

When the pasta is done, remove about 1 cup of the pasta water, then drain the pasta in a colander.

Add 3/4 cup of the reserved pasta water to the bacon drippings in the skillet and place the pan over medium heat. Bring to a boil.

Add the drained pasta to the skillet of boiling pasta water; reduce the heat to low and cook for about 1 minute. Season to taste with kosher salt and pepper.

Remove the skillet from the heat and gradually add the egg, bacon, and cheese mixture. Use tongs or a pasta fork to toss the pasta until the sauce mixture becomes silky. If necessary, add more of the hot pasta water.

Serve the hot pasta with extra Parmesan cheese and chopped Italian parsley, if desired.