In a medium pan, heat the olive oil. Sauté the onion, season with salt and pepper, cooking until translucent, about 5 minutes. Add the minced garlic and cook until fragrant, about 30 seconds. Remove from heat and set aside. 

In a small bowl, add stout beer, brown gravy packets, and Cajun seasoning. Whisk until smooth and set aside. 

In a medium sauce pot, melt butter. Add water and bring to a boil. Once water comes to a boil, turn off heat and whisk in beer gravy mixture until thickened. Incorporate the onion and garlic mixture and set aside. 

Roll out one sheet of puff pastry and place it at the bottom and press it up the sides of the baking dish. Add leftover turkey, mixed vegetables, and gravy on top of the puff pastry. 

Roll out the second sheet of puff pastry and place it on top to form the pie. Fold the pastry sheet up onto itself working around the baking dish. Slice a few small cuts into the top of the puff pastry sheet.

In a small bowl, whisk together egg and one tablespoon water. Using a pastry brush, brush the egg wash over the top of the puff pastry. 

Bake for 25 minutes, or until the puff pastry is golden brown. 

Once baked, remove from the oven and allow to set for about 10 minutes before serving. (This will help the filling to set.) Serve!