Pile the flour on a work surface and make a well. Add the eggs, egg yolk, and olive oil to the center.
Use a fork to gradually draw in the flour. Knead for 8 to 10 minutes, adding the water only if the dough feels dry, until smooth and elastic.
Wrap and rest the dough at room temperature for 30 minutes. Divide it into 2 pieces, roll each through a pasta machine until thin, and cut into spaghetti. Dust lightly with flour.
Prepare the Carbonara:
Put the guanciale in a cold skillet over medium-low heat. Cook for 8 to 10 minutes, stirring occasionally, until crisp and the fat has rendered.
Whisk the eggs, egg yolks, Pecorino, and black pepper in a bowl until thick and smooth.
Cook and Toss:
Bring a large pot of water to a boil, add the salt, and cook the fresh spaghetti for 2 to 3 minutes, until just tender. Reserve 1 cup of pasta water before draining.
Add the drained spaghetti to the skillet with the guanciale and toss to coat. Remove the skillet from the heat.
Add the egg-and-cheese mixture and toss continuously, adding pasta water a splash at a time until the sauce is glossy and creamy.
Serve immediately with extra Pecorino and black pepper.
Notes
Traditional carbonara uses guanciale, Pecorino Romano, eggs, and pepper without cream.
Keep the pan off the heat when adding the egg mixture so the sauce stays silky.
If guanciale is unavailable, use pancetta rather than smoked bacon.