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.
Off heat, add hot drained pasta. Immediately add egg mixture; toss quickly, adding pasta water a few tablespoons at a time until creamy and clinging.
Stir in bacon and serve immediately with extra cheese and black pepper.