Preheat your oven to 375°F (190°C).

In a skillet over medium heat, add 1 tablespoon of olive oil. Once hot, add the chopped mushrooms and cook for about 5 minutes until they are softened and most of the moisture has evaporated.

Add the minced garlic and chopped spinach to the skillet, cooking for an additional 2-3 minutes until the spinach is wilted. Remove from heat and let cool slightly.

In a mixing bowl, combine the cooked mushroom mixture, cream cheese, mozzarella cheese, thyme, salt, and pepper. Mix until well combined.

Using a sharp knife, create a pocket in each chicken breast by slicing horizontally, being careful not to cut all the way through.

Stuff each chicken breast with the mushroom mixture, using toothpicks to secure the opening if necessary.

In an oven-safe skillet, heat the remaining tablespoon of olive oil over medium-high heat. Sear the stuffed chicken breasts for 3-4 minutes on each side until golden brown.

Drizzle with balsamic vinegar if using, then transfer the skillet to the preheated oven. Bake for 20-25 minutes, or until the chicken is cooked through (internal temperature of 165°F).

Remove from the oven and let rest for 5 minutes before serving.