Sprinkle both sides of the chicken with salt and pepper.
Heat olive oil in a large skillet over medium-high heat. Add chicken and cook for 2 minutes per side until golden and cooked through. Remove chicken to a plate and let rest, then slice.
Add milk, chicken stock, and minced garlic to the skillet. Bring to a simmer, then add the fettuccine.
Stir the pasta every 30 seconds to prevent sticking until softened, about 3 minutes. Reduce heat to medium and continue stirring occasionally.
After 9-10 minutes, the pasta should be almost cooked with some liquid remaining. Stir in the heavy cream and grated parmesan.
Simmer for about 2 minutes, stirring, until the sauce thickens and pasta is tender. If sauce becomes too thick, add a splash of hot water to loosen.
Season with additional salt and pepper to taste. Serve immediately, garnished with fresh parsley and extra parmesan.