Preheat the oven to 375°F (190°C).
In a skillet over medium heat, melt 2 tablespoons of butter. Add the chopped mushrooms and sauté for about 5 minutes until they are soft and any liquid has evaporated.
Add the minced garlic, thyme, and parsley to the skillet. Cook for an additional 2 minutes until fragrant. Remove from heat and let cool slightly.
In a mixing bowl, combine the sautéed mushroom mixture with mozzarella and Parmesan cheese. Season with salt and pepper to taste.
Using a sharp knife, create a pocket in each chicken breast by slicing it horizontally, being careful 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 a large oven-safe skillet, heat the olive oil over medium-high heat. Season the chicken breasts with salt, pepper, and paprika (if using). Sear the chicken for 3-4 minutes on each side until golden brown.
Add the remaining 2 tablespoons of butter to the skillet, spooning it over the chicken. Transfer the skillet to the preheated oven and bake for 20-25 minutes, or until the chicken is cooked through and reaches an internal temperature of 165°F (75°C).
Remove from the oven and let the chicken rest for 5 minutes before serving.