Preheat the oven to 400 degrees F. Melt 1 tablespoon of butter in a large skillet over medium heat. Add the chicken pieces and cook until lightly browned on all sides. Remove the chicken to a plate and set aside.
Add the remaining 2 tablespoons of butter to the same skillet. Stir in the onion, carrots, and celery. Cook for 5 minutes until the vegetables soften.
Add the mushrooms and garlic. Cook for another 4 minutes until the mushrooms release their liquid and start to brown.
Sprinkle the flour over the vegetables and stir for 1 minute to cook out the raw taste. Slowly pour in the chicken broth while stirring to avoid lumps. Add the heavy cream, thyme, salt, and pepper.
Return the chicken to the skillet. Simmer the filling for 3 to 4 minutes until it thickens slightly. Remove from heat.
Transfer the filling to a 9-inch pie dish or individual ramekins. Roll out the puff pastry slightly if needed and lay it over the top. Trim excess and crimp the edges. Cut a few small slits in the center for steam.
Brush the pastry with the beaten egg. Bake for 25 to 30 minutes until the crust is deep golden and the filling bubbles at the edges.