Heat the oil in a large skillet. Add the chicken and cook 3 minutes on each side, or until tender and cooked through. Remove the chicken from the pan, let it rest for a couple of minutes, and then slice it into small strips and set aside.
Add the mushrooms and garlic to the skillet; cook for 1 minute, stirring. Sprinkle with flour, stirring quickly. Stir in the broth and milk. Reduce heat and simmer 3 minutes, until the mixture thickens- stirring occasionally. Stir in the cheeses and pepper.
Return the chicken to the skillet; cook, stirring for about 2 minutes, until the chicken warms up.
Arrange the fettuccine on four plates. Top with chicken and sauce. Garnish with parsley.