Preheat the oven to 375°F.
Heat a large skillet over medium heat and add the olive oil. Add the diced potatoes and cook until lightly browned, about 10 minutes.
Add the onions, shallots, and garlic to the skillet and cook for an additional 5 minutes. Push the potatoes to the side and add the ground venison. Cook until browned, then mix everything together. Season with tarragon, parsley, and salt.
In a separate small skillet, melt the butter and sauté the mushrooms until browned, about 6–7 minutes.
Transfer the venison and potato mixture to a pie plate. Top with the sautéed mushrooms and shredded cheese.
Carefully place the pie crust over the filling and pinch the edges around the pie plate. Cut a few small holes in the top to allow steam to escape. Lightly brush the crust with egg wash.
Bake the meat pie for 40–45 minutes, or until the crust is golden brown. Let cool slightly before serving.