Bring a large pot of salted water to a boil. Add the fettuccine and cook according to package directions until al dente. Reserve about a cup of the pasta water, then drain.
While the pasta cooks, pat the chicken dry and season with a little salt and pepper. Heat the olive oil in a skillet over medium-high heat. Cook the chicken until golden and cooked through, about 6–7 minutes per side depending on thickness. Transfer to a cutting board and let rest for a few minutes before slicing into bite-sized pieces.
In the same skillet, melt the butter over medium heat. Add the minced garlic and cook for about 30 seconds until fragrant, being careful not to burn it.
Pour in the heavy cream and bring to a simmer. Reduce the heat to low and whisk in the Parmesan cheese until the sauce thickens slightly. Season with salt and pepper to taste.
Return the sliced chicken to the pan and simmer gently for another minute to warm through. If the sauce seems thick, add a small splash of the reserved pasta water to loosen it to a silky consistency.
Toss the drained pasta into the sauce, coating it evenly. If needed, add more pasta water a little at a time to reach your desired sauce coats-the-pasta consistency.
Garnish with chopped parsley (if using) and serve immediately. A light extra sprinkle of Parmesan on top makes it extra comforting.