Preheat the oven to 425 degrees F. Place one sheet of puff pastry into the 9-inch pie dish. Place in the fridge.
Meanwhile, preheat a medium pan with a tablespoon of oil. Add chicken and cook until golden and cooked through. Set aside on a plate.
Add remaining oil, onion and cook for about 5 minutes. Add mushrooms and cook until the water evaporates and they begin to brown.
Add garlic and butter and stir in the flour. Add cooked chicken and whisk in broth and milk. Season with salt and pepper. Cook until the mixture is smooth and thickened.
Transfer the mixture to your pie dish. Cover with the remaining sheet of puff pastry and line up the corners with the corners of the bottom puff pastry sheet. Brush whisked egg all over the top crust.
Bake for 25-30 minutes until the crust is puffed and golden brown. Cool slightly before serving.