Preheat your oven to 375°F (190°C) and lightly grease a 9x13 inch baking dish with olive oil or non-stick cooking spray.
Make the stuffing by melting 2 tablespoons of butter in a large skillet over medium heat, then add the chopped onion and cook for 3 minutes until softened before stirring in the minced garlic and cooking for 1 more minute until fragrant.
Add the finely chopped mushrooms to the skillet, season with a pinch of salt and pepper, and cook for 5 to 7 minutes until they have released their moisture and turned a deep golden brown, then pour in the white wine to deglaze the pan, scraping up any browned bits, and cook for 1 minute until the liquid evaporates.
Transfer the mushroom mixture to a medium bowl and let it cool for 5 minutes, then stir in the softened cream cheese, grated Parmesan, and fresh parsley until the mixture is fully combined and creamy.
Use a sharp paring knife to cut a deep horizontal pocket into the thickest side of each chicken breast, being careful not to cut all the way through to the other side.
Divide the mushroom stuffing evenly among the 4 chicken breasts, using a spoon to press about 2 to 3 tablespoons of the mixture firmly into each pocket, then secure the openings closed with kitchen twine or several toothpicks.
Place the stuffed chicken breasts in the prepared baking dish, rub the outsides with 1 tablespoon of olive oil, and season all over with 1/2 teaspoon salt and 1/4 teaspoon pepper.
Bake for 25 to 30 minutes, or until the chicken is cooked through and registers 165°F on an instant-read thermometer inserted into the thickest part of the meat, then let it rest for 5 minutes before removing any twine or toothpicks and serving.