Heat the oven to 375°F and lightly grease a 9-by-13-inch baking dish.

Pat the chicken dry and season it with about 3/4 teaspoon of the kosher salt and all of the black pepper.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Brown the chicken in batches for about 2–3 minutes per side, then transfer it to the prepared baking dish; it does not need to be fully cooked.

Add the remaining olive oil and sliced mushrooms to the skillet. Cook for 6–8 minutes, stirring occasionally, until the mushrooms release their moisture and become nicely browned.

Reduce the heat to medium and add 2 tablespoons butter with the minced garlic. Cook for 30–60 seconds, just until the garlic is fragrant.

Stir in the remaining 2 tablespoons butter. Sprinkle in the flour and cook for 1 minute, stirring constantly to form the base of the sauce.

Gradually whisk in the chicken broth, followed by the heavy cream. Add the Italian seasoning, remaining salt, and optional red pepper flakes, then simmer for 3–5 minutes until slightly thickened.

Remove the skillet from the heat and stir in the 1/2 cup Parmesan. Adding the cheese off the heat helps it melt into a smooth sauce.

Pour the mushroom-Parmesan sauce over the chicken in the baking dish. Sprinkle the top with the optional mozzarella and the additional 2 tablespoons Parmesan.

Bake the casserole uncovered for 25–30 minutes, until bubbling and the chicken reaches 165°F in the thickest pieces.

Let the casserole rest for 5–10 minutes before garnishing with parsley. Serve it with pasta, rice, mashed potatoes, or crusty bread.