Heat 2 teaspoons oil in a large skillet over medium-high heat until shimmering.
Season the chicken breasts lightly with pepper, add to the skillet, and cook about 3 minutes per side until cooked through; remove to a plate and let rest a few minutes, then slice into strips.
Add the sliced mushrooms and minced garlic to the skillet and cook about 1 minute, stirring, until the mushrooms begin to soften.
Sprinkle 2 teaspoons flour over the mushrooms and garlic, stirring quickly to combine and cook briefly.
Slowly pour in 1 cup chicken broth and 1/2 cup evaporated milk while stirring; reduce heat and simmer about 3 minutes, stirring occasionally, until the sauce thickens slightly.
Stir in 4 tablespoons 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 about 2 minutes, stirring, until the chicken is heated through and coated with sauce.
Divide 2 cups hot cooked fettuccine among plates, top with the chicken and mushroom sauce, and garnish with chopped parsley if desired.
