In a large skillet over medium-high heat, heat oil. Add chicken; season with salt and pepper. Cook, turning occasionally, until golden brown and cooked through, about 8 minutes per side. Transfer to a cutting board. Let rest 10 minutes, then slice.

In same skillet over medium heat, combine broth, milk, and garlic; season with salt and pepper. Bring to a simmer. Add fettuccine and cook, tossing occasionally, until barely al dente, about 10 minutes.

Stir in Parmesan and cream. Bring to a simmer and cook, stirring frequently, until sauce thickens and pasta is al dente, 2 to 3 minutes.

Remove from heat and stir in chicken. Top with parsley.