Bring a large pot of generously salted water to a boil. Cook the pasta according to the package directions.
While the pasta is cooking, crack the eggs into a large, shallow serving bowl. Whisk to break up the egg yolks. Stir in the cheeses.
Also while the pasta is cooking, heat the olive oil in a skillet over medium heat. Add the pancetta and garlic. Cook until the pancetta is crispy. Discard the garlic.
As soon as the pasta is done, drain it in a colander. Shake a few times to remove the excess water, then put it in the bowl with the eggs and cheese, and toss. The hot pasta will cook the eggs.
Add the pancetta with the olive oil to the pasta. Stir in the salt and garnish with the basil. Sprinkle the pepper on top. Serve immediately with additional Parmesan.