Heat olive oil in a large skillet over medium-high heat. Add the chicken and season with salt and pepper. Cook, turning occasionally, until golden brown and cooked through, about 8 minutes per side, or until the internal temperature reaches 165°F.

Transfer the chicken to a cutting board and let it rest for 10 minutes before slicing.

In the same skillet, add the garlic, chicken broth, and milk. Season lightly with salt and pepper. Bring to a simmer over medium heat.

Add the fettuccine and cook, stirring frequently, until the sauce begins to reduce slightly, about 3 minutes. Continue cooking, tossing occasionally, until the pasta is al dente, about 8 to 10 minutes more.

Stir in the grated Parmesan and heavy cream until smooth and combined. Continue cooking, stirring occasionally, until the sauce thickens, about 3 more minutes.

Remove from heat and stir in the sliced chicken. Sprinkle with chopped parsley before serving.