Preheat your oven to 375°F (190°C).
In a skillet over medium heat, melt 1 tablespoon of butter. Add the chopped mushrooms and cook for about 5 minutes, until they are softened and any liquid has evaporated.
Add the minced garlic to the skillet and cook for an additional 1-2 minutes until fragrant. Remove from heat and let cool slightly.
In a mixing bowl, combine the cooked mushrooms and garlic with cream cheese, mozzarella cheese, salt, pepper, thyme, and paprika. Mix until well combined.
Carefully slice a pocket into each chicken breast, making sure not to cut all the way through.
Stuff each chicken breast with the mushroom and cheese mixture, securing the opening with toothpicks if necessary.
In the same skillet, melt the remaining 2 tablespoons of butter over medium heat. Add the stuffed chicken breasts and sear for about 3-4 minutes on each side until golden brown.
Transfer the seared chicken to a baking dish and bake in the preheated oven for 20-25 minutes, or until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).
Remove from the oven, let rest for 5 minutes, and garnish with fresh parsley before serving.