Heat 2 teaspoons oil in a large skillet over medium-high heat.
Add the chicken breasts and cook about 3 minutes per side, or until cooked through; transfer to a plate and let rest for a couple of minutes, then slice into strips.
Add the sliced mushrooms and minced garlic to the skillet and cook about 1 minute, stirring, until mushrooms begin to soften.
Sprinkle 2 teaspoons flour over the mushrooms and garlic and stir quickly to combine.
Stir in 1 cup chicken broth and 1/2 cup evaporated milk, reduce heat and simmer about 3 minutes, stirring occasionally, until sauce thickens.
Turn off the heat and stir in the whipped cream cheese and 1 1/2 cups grated Parmesan until smooth; season with freshly ground black pepper to taste.
Return the sliced chicken to the skillet and cook, stirring, about 2 minutes until heated through and coated with sauce.
Divide 2 cups hot cooked fettuccine among four plates, top with the chicken and mushroom sauce, and garnish with chopped parsley if desired.