Season chicken with salt and pepper.
Heat 3 tablespoons oil over in large skillet (use a skillet large enough to fit the pasta) over medium-high heat. Add chicken and cook for 3 - 5 minutes until lightly golden and cooked through.
Add the mushrooms and sauté until the liquid from the mushrooms evaporates and the mushrooms become pale golden, about 2-3 minutes. Transfer chicken and mushrooms mixture to a plate. Reserve.
Add the remaining tablespoon of oil and the butter to the same skillet. Add the onion and garlic, and sauté until the onion is translucent, about 5 minutes. Add the wine and simmer until it evaporates, about 2 minutes.
Add milk and chicken stock to the skillet; bring to boil. Then add pasta and mix to combine. Cover, reduce heat to medium and cook for 12 minutes, stirring occasionally, or until the pasta is tender.
Add cream, parmesan and cream cheese, if using, and mix. Simmer, stirring occasionally, for about 2 minutes, until sauce is thickened. Add parsley and mix to combine.
Sprinkle the mozzarella cheese over the pasta. Bake, uncovered, until golden brown on top and the sauce bubbles, about 20 minutes.