To make the filling, sauté the meats on high heat in a Dutch oven until well browned. Add onion, shallots, and garlic; sauté until soft. Pour in red wine to deglaze the pan. Add the venison or beef stock and potatoes; reduce heat to simmer. Fold in spices and fresh thyme. Cook until no liquid remains and the potatoes are fork tender, about 20 minutes. Mash the potatoes into the meat mixture. Season with salt and pepper. Transfer to a bowl, cover, and refrigerate overnight.

To ready the pie dough, preheat the oven to 350°. Whisk egg and 2 teaspoons water for egg wash. Roll out pie dough and place into a 10-inch pie plate or rectangular tart pan. Brush lightly with half of egg wash.

Fill prepared crust with chilled meat mixture and cover with the other pie crust on top. Brush top crust with rest of egg wash (you may not need to use it all). Crimp edges and cut air vents to allow steam to escape.

Bake until golden brown, about 45 minutes. Remove and allow to cool for 15 minutes before serving.