Prepare the pudding: Whisk the instant vanilla pudding mix with the cold milk in a large bowl until thickened, about 2 minutes. Gently fold in the thawed whipped topping. Set aside.
Prepare the chocolate glaze: In a small saucepan, combine the water, sugar, and cocoa powder. Bring the mixture to a boil over medium heat, stirring constantly. Remove from heat and stir in the butter and vanilla extract until the butter melts completely. Let the glaze cool slightly, about 10 minutes.
Assemble the first layer: Arrange a single layer of graham crackers on the bottom of a 9×13 inch baking dish. Break crackers if necessary to cover the bottom completely.
Add the filling: Spread half of the pudding mixture evenly over the graham cracker layer.
Add the second layer: Top the pudding layer with another layer of graham crackers.
Add the remaining filling: Spread the remaining half of the pudding mixture over the second cracker layer.
Add the final layer: Top with a final layer of graham crackers.
Glaze the cake: Pour the slightly cooled chocolate glaze evenly over the top layer of graham crackers, spreading it to cover the edges.
Chill: Cover the dish with plastic wrap and refrigerate for at least 6 hours, or preferably overnight, to allow the graham crackers to soften.
Notes
You can use chocolate instant pudding instead of vanilla for a richer chocolate flavor throughout the filling.
For easier slicing, run a knife under hot water and wipe it dry between cuts.
This dessert tastes best after chilling overnight, as the crackers fully absorb moisture.