Use a wide, deep pan that fits the pasta flat and leaves room for stirring.
Season both sides of the chicken with salt and pepper.
Heat the oil over medium-high heat. Sear the chicken for 2 minutes per side until golden and cooked through. Transfer it to a plate to rest for 5 minutes, then slice.
Add the milk, chicken stock, and garlic to the pan. Bring to a simmer, then add the fettuccine.
Move the pasta every 30 seconds until softened, about 3 minutes, then reduce the heat to medium and stir every couple of minutes.
At 9 to 10 minutes, when the pasta is nearly tender and liquid still covers the pan base, stir in the cream and Parmesan.
Simmer and stir for about 2 minutes until the sauce thickens and the pasta is cooked. Add a splash of hot water if it becomes too thick.
Taste for seasoning, return the sliced chicken to the pan, and serve immediately with parsley and Parmesan.
Notes
For a silky sauce, keep the pasta moving and take the pan off the heat as soon as the noodles are tender.