Heat 2 teaspoons oil in a large skillet over medium-high heat.
Add the chicken breasts and cook about 3 minutes per side, until cooked through; remove to a plate and let rest a couple minutes, then slice into strips.
Add the sliced mushrooms and minced garlic to the skillet and cook about 1 minute, stirring, until softened.
Sprinkle 2 teaspoons flour over the mushrooms and garlic and stir quickly to coat.
Slowly pour in 1 cup chicken broth and 1/2 cup evaporated milk, stirring to combine; reduce heat and simmer about 3 minutes until slightly thickened, stirring occasionally.
Stir in 4 tablespoons whipped cream cheese, 1 1/2 cups grated Parmesan, and freshly ground black pepper to taste until the cheeses melt and the sauce is smooth.
Return the sliced chicken to the skillet and cook about 2 minutes, stirring, until warmed through and coated in the sauce.
Divide 2 cups hot cooked fettuccine among four plates, top with the chicken and sauce, and garnish with chopped parsley if desired.