Cook the fettuccine or spaghetti according to package instructions until al dente. Drain and set aside.
While the pasta cooks, season the chicken breasts with salt and pepper. In a large skillet, heat the olive oil over medium heat. Add the chicken and cook for about 6-7 minutes on each side, or until cooked through and no longer pink in the center. Remove the chicken from the skillet and let it rest for a few minutes before slicing it into strips.
In the same skillet, add the minced garlic and sauté for about 30 seconds until fragrant.
Pour in the chicken broth and bring to a simmer, scraping any browned bits from the bottom of the skillet.
Stir in the heavy cream, Parmesan cheese, lemon juice, and lemon zest. Cook for 2-3 minutes until the sauce thickens slightly.
Add the cooked pasta to the skillet and toss to coat in the sauce. Add the sliced chicken on top and gently mix everything together.
Serve immediately, garnished with fresh parsley.