Heat the oven to 375°F (190°C) and lightly grease a 9-by-13-inch baking dish.

Pat the chicken dry and season both sides with the kosher salt and black pepper. If the breasts are very thick, cut them horizontally into cutlets so they cook evenly.

Heat the olive oil and 1 tablespoon of butter in a large oven-safe skillet. Brown the chicken for about 2 to 3 minutes per side, then transfer it to the prepared baking dish or a plate; it does not need to be fully cooked yet.

Add the remaining tablespoon of butter to the skillet. Cook the mushrooms and onion for 5 to 7 minutes, stirring occasionally, until the mushrooms release their moisture and begin to brown.

Add the minced garlic and Italian seasoning. Cook for 30 to 60 seconds, just until the garlic is fragrant and the herbs are aromatic.

Pour in the chicken broth and scrape up the browned bits from the bottom of the pan. Stir in the heavy cream and sour cream or softened cream cheese, then simmer gently for 2 to 3 minutes without allowing the sauce to boil vigorously.

Stir in the Parmesan cheese and taste the sauce carefully. Add more salt or pepper if needed, remembering that the cheeses will contribute additional savory flavor.

Return the chicken to the sauce if using an oven-safe skillet, or arrange it in the greased casserole dish and pour the mushroom sauce over it. Sprinkle the shredded mozzarella evenly over the top.

Bake the casserole uncovered for 20 to 25 minutes. Check the thickest chicken piece with an instant-read thermometer and continue cooking only until it reaches 165°F (74°C) internally and the sauce is bubbling.

Let the casserole rest for 5 minutes before serving. Finish with the chopped fresh parsley and spoon the creamy sauce over each portion.