Warm the olive oil in a large skillet over medium-high heat. Season the chicken breasts generously with kosher salt and freshly ground black pepper. Add the chicken to the skillet and cook, turning occasionally, until both sides are golden brown and the meat is fully cooked, approximately 8 minutes per side. Transfer the chicken to a cutting board and allow it to rest for 10 minutes before slicing.

In the same skillet, over medium heat, combine the chicken broth, whole milk, and finely chopped garlic. Season with salt and pepper to taste. Bring the mixture to a gentle simmer. Add the fettuccine to the skillet and cook, stirring occasionally, until the pasta is just shy of al dente, around 10 minutes.

Incorporate the finely grated Parmesan cheese and heavy cream into the skillet. Return the mixture to a simmer, stirring frequently until the sauce thickens and the pasta reaches al dente perfection, which should take about 2 to 3 minutes.

Remove the skillet from heat and stir in the sliced chicken. Garnish with chopped fresh parsley before serving.