Bring a large pot of water to a boil. Cook the spaghetti according to the package directions until al dente. Take out about ½ cup of the pasta water, then drain the cooked pasta and set it aside.
In a large frying pan or sauté pan over medium-high heat, cook the bacon until it becomes crispy, about 5–7 minutes. Remove the cooked bacon and place it on a paper towel-lined plate. Leave about 1–2 tablespoons of bacon fat in the skillet.
In the same skillet, add the bite-sized pieces of chicken and garlic. Cook them over medium heat until the chicken is golden brown and fully cooked, about 6–8 minutes. Transfer the tender chicken to a plate and set it aside.
In a large bowl, whisk together the eggs, egg yolks, Parmesan cheese, heavy cream, salt, and black pepper until well combined. This cream mixture will create a beautiful silky sauce for your chicken carbonara recipe.
Reduce the heat to medium-low. Place the pasta into the skillet. Slowly pour in the egg-based sauce, tossing quickly to coat the pasta without scrambling the eggs. Add the reserved pasta water, a little at a time, until the creamy sauce reaches your desired consistency.
Stir in the cooked chicken and bacon, and toss everything together. Remove it from the heat and serve it immediately, garnished with chopped parsley and extra Parmesan cheese.