Step 1: Preheat your oven to 375°F (190°C). Preparing your oven ahead of time ensures that it reaches the correct temperature, allowing for even cooking. Position a rack in the center of the oven for optimal heat distribution.

Step 2: Carefully slice each chicken breast horizontally to create a pocket for the filling. Avoid cutting all the way through; you want to create a pouch that can be filled. The larger the pocket, the more filling you can add, enhancing the flavor and texture of the dish.

Step 3: In a skillet over medium heat, melt 1 tablespoon of butter. Add the minced garlic and cook for about 30 seconds until fragrant. Be careful not to burn the garlic, as this can impart a bitter taste. Add the finely chopped mushrooms and cook until they soften and release their moisture, about 5-7 minutes. Stir frequently to ensure even cooking.

Step 4: Stir in the softened cream cheese and shredded mozzarella cheese into the mushroom mixture. Continue stirring until the cheeses are melted and everything is well combined. Season with salt and pepper to taste. This mixture should be creamy and flavorful. If it seems too thick, you can add a tablespoon of chicken broth to thin it out.

Step 5: Fill each chicken breast pocket with the mushroom-cheese mixture. Use a spoon to carefully pack the filling into the pocket, ensuring it's evenly distributed. Secure the opening of each chicken breast with toothpicks to prevent the filling from leaking out during cooking. Insert the toothpicks at an angle to hold the edges together tightly.

Step 6: In the same skillet, melt the remaining tablespoon of butter over medium-high heat. Sear the stuffed chicken breasts on both sides for about 2-3 minutes per side, until they are golden brown. Searing adds color and flavor to the chicken and helps to seal in the juices.

Step 7: Pour the chicken broth into the skillet around the chicken breasts. The broth will keep the chicken moist and add flavor as it bakes. Cover the skillet tightly with a lid or aluminum foil.

Step 8: Transfer the skillet to the preheated oven and bake for 25 minutes, or until the chicken is cooked through. The internal temperature of the chicken should reach 165°F (74°C). Use a meat thermometer to check the temperature at the thickest part of the chicken breast.

Step 9: Remove the skillet from the oven and let the chicken rest for a few minutes before serving. Garnish with fresh thyme sprigs for added flavor and visual appeal. Remove the toothpicks before serving.