Preheat the oven to 415 degrees F (210 degrees C). Line a baking sheet with parchment paper.

Arrange pie crust on the prepared baking sheet. Mix walnuts, brown sugar, 1 egg, butter, and vanilla extract together in a bowl. Spread the walnut filling over the pie crust, leaving a 1-inch border. Reserve 2 tablespoons of the filling in the bowl.

Arrange apple slices in a circular pattern over the filling. Sprinkle cranberries over the apples. Drop small amounts of the reserved walnut filling here and there over the apples. Fold crust over the filling, overlapping as necessary to form a rustic circle.

Beat remaining egg and water together to make egg wash. Brush edges of crust with the egg wash. Sprinkle Demerara sugar on top.

Bake in the preheated oven until the crust is browned and apples are bubbling, about 30 minutes. Place baking sheet on a cooling rack for 30 minutes. Lift up tart using the parchment paper and place directly on a cooling rack for 1 hour.

Run a large knife or angled cake spatula under the tart to release the parchment paper and transfer to a serving tray.