Preheat the oven to 400°F. Line a baking sheet with parchment paper.
Season the chicken breasts with salt and pepper. In a large skillet, heat olive oil over medium heat. Add the chicken and cook for about 5-7 minutes on each side, or until cooked through. Remove from the skillet and let cool.
In the same skillet, add the mushrooms, onion, and garlic. Sauté for about 5 minutes until the vegetables are softened and the moisture from the mushrooms has evaporated. Stir in thyme and cook for an additional minute. Remove from heat and let cool slightly.
In a bowl, combine the cream cheese and parsley. Stir in the mushroom mixture until well combined.
Once the chicken has cooled, slice it in half horizontally to create thinner pieces. Place a portion of the mushroom mixture on each piece of chicken, then fold the chicken over to enclose the filling.
Roll out the puff pastry on a lightly floured surface to about 1/8 inch thick. Cut the pastry into rectangles large enough to wrap around each chicken piece.
Place the stuffed chicken in the center of each pastry rectangle. Fold the pastry over the chicken and seal the edges by pressing with a fork. Place seam-side down on the prepared baking sheet.
Brush the tops of the pastry with the beaten egg. Cut a few slits in the top of each pastry to allow steam to escape.
Bake in the preheated oven for 25-30 minutes, or until the pastry is golden brown and flaky.
Let cool for a few minutes before serving.
Notes
For added flavor, you can include spinach or kale in the mushroom mixture.
Serve with a side salad or steamed vegetables for a complete meal.