Season the chicken with the salt and pepper.
Melt the butter in a deep 12-inch skillet over medium-high heat. Add the chicken and cook for 5–6 minutes, stirring now and then, until browned and cooked through.
Add the garlic and cook for 30 seconds.
Pour in the broth and milk and stir in the pasta. Bring just to a boil, then reduce the heat to medium-low.
Simmer uncovered for 12–14 minutes, stirring often, until the pasta is tender and the sauce has thickened. Add a splash of broth if the pan gets dry before the pasta is done.
Remove from the heat and stir in the Parmesan until melted. Let it sit for 2 minutes, then serve.