Preheat the oven to 400°F (200°C) and let the puff pastry thaw according to package instructions.
In a large skillet, heat 1 tablespoon olive oil over medium-high heat. Add the 16 oz diced chicken breast and cook for 5 to 7 minutes until browned. Remove and set aside.
In the same skillet, melt 2 tablespoons of butter over medium heat. Add the chopped onion and sauté for 3 to 4 minutes. Stir in the minced garlic and sliced mushrooms, cooking for 5 more minutes until tender.
Melt the remaining 2 tablespoons of butter with the vegetables. Sprinkle 3 tablespoons all-purpose flour over the top, stirring constantly for 1 minute.
Gradually whisk in 1 cup chicken broth. Simmer for 2 to 3 minutes until thickened.
Reduce heat to low. Stir in 1/2 cup heavy cream, 1 teaspoon fresh thyme, and the cooked chicken. Season with salt and black pepper to taste. Remove from heat.
Pour the filling into a 9-inch baking dish.
Unroll the thawed puff pastry sheet over the dish, trimming edges. Cut 2 to 3 small slits or an X in the center of the pastry for venting [2.5].
Brush the pastry top evenly with the beaten egg.
Bake at 400°F (200°C) for 20 to 25 minutes until puffed and golden. Let cool for 5 to 10 minutes before serving.