Cook the fettuccine pasta according to package instructions until al dente. Drain and set aside, reserving 1/2 cup of the pasta water.
In a large skillet, heat the olive oil over medium-high heat. Add the chicken pieces, season with salt, pepper, and red pepper flakes (if using), and cook until browned and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.
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 zest, and lemon juice. Let the sauce simmer for 2-3 minutes until slightly thickened.
Return the cooked chicken to the skillet, along with the drained pasta. Toss everything together, adding reserved pasta water as needed to reach desired consistency.
Remove from heat and stir in the chopped parsley. Serve immediately, garnished with additional Parmesan cheese if desired.