Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season the chicken pieces with salt and pepper. Add the chicken to the pot in a single layer and cook for 5-7 minutes, turning occasionally, until browned on all sides and cooked through. The internal temperature should reach 165°F (74°C). Transfer the cooked chicken to a plate and set aside.
Reduce the heat to medium and add the butter to the same pot. Once it melts, add the minced garlic and cook for about 30-60 seconds until fragrant, stirring continuously to prevent it from browning.
Pour in the chicken broth and heavy cream, scraping up any browned bits from the bottom of the pot. Add the broken fettuccine and stir to submerge it in the liquid.
Bring the liquid to a boil, then reduce the heat to maintain a steady simmer. Cook for about 10 minutes, stirring occasionally to prevent sticking, until the sauce begins to thicken.
Stir in the Parmesan cheese, lemon zest, lemon juice, capers, and olives. Continue to simmer, stirring frequently, for another 5-8 minutes, or until the pasta is al dente and the sauce has thickened to your liking.
Return the cooked chicken and any accumulated juices to the pot. Stir to combine and cook for 1-2 minutes, just until the chicken is warmed through.
Remove from the heat and serve immediately. Garnish each serving with fresh parsley and extra Parmesan cheese, if desired.