Bring a large pot of salted water to a boil.
Whisk eggs, 3/4 cup Parmesan, and black pepper in a bowl until smooth.
Boil spaghetti until al dente. Reserve 3/4 cup cooking water, then drain.
Crisp bacon in a large skillet over medium heat. Transfer to a plate.
Add olive oil to the skillet. Sear chicken in bacon drippings until cooked through.
Reduce heat to low and sauté garlic for 30 seconds, just until fragrant.
Add drained pasta, bacon, and chicken to the pan. Toss with a splash of hot pasta water.
Remove from heat. Quickly stir in the egg mixture, tossing until glossy and creamy.
Loosen with more pasta water as needed. Finish with parsley and remaining Parmesan. Serve immediately.