Preheat the oven to 400 (200 C) degrees and line a large tray with paper towels.
Arrange the bacon on a rimmed baking sheet. Bake for 15-20 minutes, flipping the pieces halfway through, until the bacon is deeply colored and crisp. Transfer to the tray to absorb excess fat. Chop the bacon into small pieces.
Whisk the egg yolks, whole egg, Parmesan and Pecorino cheeses, ¼ teaspoon salt and black pepper in a large mixing bowl. If you have a large pasta serving bowl, use that to mix and serve.
Bring 4-5 quarts of water to a boil in a large saucepan or pot and add 1 tablespoon salt. Add the pasta and cook according to package directions until al dente (usually about 10 minutes.) Scoop out 3/4 cup of the cooking water before draining the pasta.
Whisk ½ cup of the hot water into into the egg yolk mixture — this tempers the eggs so they won't scramble.
Immediately dump the hot pasta into the bowl, then toss it all together quickly and thoroughly, using tongs or two spoons. You can add more water if the sauce seems dry. Sprinkle the bacon and parsley over the pasta and toss to combine. Serve right away, with additional grated cheese on the side.
Add a small grated garlic clove and a pinch of crushed red pepper to the eggs.
Toss a tablespoon or two of soft butter to the pasta (it's not traditional but very delicious).
Throw in 1 1/2 cups frozen peas to the pasta during the last 2 minutes of cooking.
If you'd like to use pancetta instead of bacon, bake 1/2-inch cubes 5-10 minutes at 400 degrees until crisp.