Cook spaghetti according to package directions. Reserve 1 cup pasta water before draining.
In a bowl, whisk together egg yolks, whole egg, cheese, and black pepper. Set aside.
Cook bacon in a large skillet over medium heat until crispy. Remove bacon but leave drippings in the pan.
Add olive oil and garlic to the same pan, then cook chicken slices until cooked through, about 5-6 minutes. Season with salt.
Remove pan from heat and add hot drained pasta to the skillet. Immediately pour egg mixture over pasta and toss quickly, adding pasta water a few tablespoons at a time until sauce is creamy and clings to the noodles.
Stir in bacon and serve immediately with extra cheese and black pepper.