Preheat your oven to 375°F (190°C).
Butterfly the chicken breasts, cutting them horizontally, but not all the way through, creating a pocket for the filling.
In a skillet, heat olive oil over medium heat. Add garlic and mushrooms, and sauté for 5-7 minutes until mushrooms are soft and any moisture evaporates.
Remove from heat and mix in mozzarella, Parmesan, parsley, oregano, salt, and pepper.
Stuff each chicken breast with the mushroom mixture, then secure with toothpicks if necessary.
Heat a skillet over medium-high heat and sear both sides of the chicken breasts until golden brown (about 2-3 minutes per side).
Transfer the chicken to a baking dish and add a small pat of butter on top of each breast.
Bake for 20-25 minutes, until the chicken reaches an internal temperature of 165°F (74°C).
Let the chicken rest for 5 minutes before serving.