In a large skillet over medium-high heat, heat oil. Add chicken; season with salt and pepper. Cook, turning occasionally, until golden brown and an instant-read thermometer inserted into thickest part registers 165°, about 8 minutes per side.
Transfer chicken to a cutting board. Let rest 10 minutes, then slice.
Add garlic, broth, and milk to skillet; season with salt and pepper. Bring to a simmer over medium heat. Add fettuccine and cook, stirring frequently, until sauce is slightly reduced, about 3 minutes. Continue to cook, tossing occasionally, until pasta is al dente, 8 to 10 minutes more.
Stir in Parmesan and cream until combined and smooth. Continue to cook, stirring occasionally, until sauce is thickened, about 3 minutes more.
Remove from heat and stir in sliced chicken. Top with parsley.
