Bring a large pot of salted water to a boil. Add the spaghetti and cook until al dente. Save 1 cup of the pasta water before draining.
While the pasta cooks, cook the pancetta in a large skillet over medium heat until crisp, about 4-5 minutes. Stir in the garlic and cook for another 30 seconds. Remove from heat.
In a bowl, whisk together the egg yolks, whole egg, and Parmesan cheese until smooth.
Quickly add the hot, drained pasta to the skillet with pancetta and garlic. Toss to combine.
Slowly pour in the egg mixture while stirring constantly, allowing the heat of the pasta to cook the eggs without scrambling them.
Add a little reserved pasta water if needed to create a smooth, creamy sauce.
Season with salt and freshly cracked black pepper. Serve immediately with extra Parmesan.