Preheat your oven to 375°F (190°C).
In a skillet over medium heat, melt the butter. Add the chopped mushrooms, garlic, thyme, salt, and pepper. Sauté for about 5-7 minutes until the mushrooms are tender and the liquid has evaporated. Remove from heat and let cool slightly.
In a mixing bowl, combine the sautéed mushroom mixture with cream cheese, mozzarella cheese, and 1/4 cup Parmesan cheese. Mix until well combined.
Carefully slice a pocket into each chicken breast, making sure not to cut all the way through. Stuff each pocket with the mushroom and cheese mixture.
Heat olive oil in an oven-safe skillet over medium-high heat. Season the outside of the chicken breasts with salt and pepper. Sear the stuffed chicken breasts for about 3-4 minutes on each side until golden brown.
Remove the chicken from the skillet and set aside. In the same skillet, add chicken broth and heavy cream, scraping up any browned bits from the bottom. Stir in the remaining 1/4 cup Parmesan cheese and bring to a simmer.
Return the stuffed chicken to the skillet, spoon some sauce over the top, and transfer the skillet to the preheated oven. Bake for 20-25 minutes or until the chicken is cooked through (internal temperature should reach 165°F).
Once cooked, remove from the oven and let rest for 5 minutes. Garnish with fresh parsley before serving.