Bring a large pot of salted water to a rolling boil. Add the broken spaghetti or linguine and cook until just al dente, usually about 8-10 minutes. Drain and set aside.

In a large skillet, heat 2 tablespoons of olive oil or butter over medium heat. Add the finely chopped onion and sliced mushrooms. Sauté for 6–8 minutes until browned and tender. Add minced garlic and cook for an additional minute.

Sprinkle ¼ cup of flour over the vegetables and stir for 1 minute. Gradually whisk in 2 cups of chicken broth and 1 cup of milk, and the optional wine. Cook and stir for about 5 minutes until the sauce thickens.

Reduce heat to low and stir in sour cream, Parmesan cheese, mozzarella cheese, salt, pepper, and thyme or Italian seasoning. Mix until smooth.

In a large bowl, combine the cooked pasta and shredded or diced chicken with the creamy sauce. Toss carefully until evenly coated.

Transfer the mixture to a greased 9x13-inch baking dish, spreading it evenly. Sprinkle with remaining mozzarella cheese on top.

Preheat oven to 350°F (175°C). Bake covered for 20 minutes, then remove the foil and bake for an additional 10 minutes until golden and bubbly.

Let rest for 5–10 minutes before serving. Garnish with chopped fresh parsley.