The fettuccine cooks in the milk and broth rather than a separate pot of water, so the starch it sheds thickens the sauce instead of going down the drain. One skillet, and the chicken rests while it happens.
Heat the oil in a large skillet over medium-high. Season the chicken with salt and pepper, lay it in, and turn it now and then until both sides are golden and the thickest part reads 165°F — about 8 minutes a side.
Move the chicken to a board and leave it there 10 minutes before slicing. Cut it straight away and the juice ends up on the board instead of back in the meat.
Pour the broth and milk into the same skillet, add the garlic, and season again. Bring it to a simmer over medium heat, then add the fettuccine dry. Stir frequently for the first 3 minutes or so, until the liquid has reduced a little, then toss occasionally for another 8 to 10 until the pasta is al dente. It cooks in the sauce rather than in a separate pot of water, which is the whole point — the starch it sheds is what thickens everything.
Stir in the Parmesan and cream until smooth, bring it back to a simmer, and cook about 3 minutes more, tossing, until the sauce clings to the pasta instead of pooling underneath.
Off the heat, stir the sliced chicken back in. Scatter the parsley over the top.
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