Cook the spaghetti in a large pot of salted boiling water until al dente. Drain, reserving 1/2 cup of the pasta water, and set aside.
In a large skillet, cook the bacon over medium heat until crispy. Remove the bacon and set aside, leaving the bacon fat in the skillet.
Add the chicken pieces to the skillet and season with salt and pepper. Cook until the chicken is golden and cooked through, about 5 to 7 minutes. Add the minced garlic in the last minute of cooking.
In a bowl, whisk together the eggs and grated Parmesan cheese. Slowly pour in the reserved pasta water, while whisking, to temper the eggs.
Lower the heat under the skillet and add the cooked spaghetti. Pour the egg and cheese mixture over the spaghetti, tossing continuously until the pasta is coated and the sauce has thickened, about 2 to 3 minutes.
Return the cooked bacon to the skillet, toss everything together, and season with additional salt and pepper to taste.
Serve immediately, garnished with extra Parmesan cheese if desired.