In a large mixing bowl, whisk together the instant vanilla pudding mix and cold milk until well combined. Allow it to sit for about 2 minutes to thicken.
Gently fold the whipped topping into the pudding mixture until smooth and well combined.
In the bottom of a 9x13-inch baking dish, layer half of the graham crackers to cover the base. Break crackers as necessary to fit.
Spread half of the pudding mixture evenly over the graham crackers.
Top the pudding layer with another layer of graham crackers.
Spread the remaining pudding mixture over the second layer of graham crackers.
Finish with a final layer of graham crackers.
In a separate bowl, combine the melted butter, powdered sugar, cocoa powder, milk, and vanilla extract. Whisk until smooth to make the chocolate glaze.
Pour the chocolate glaze over the top layer of graham crackers, spreading it evenly to cover.
Cover the dish with plastic wrap and refrigerate for at least 4 hours, preferably overnight, to allow the flavors to meld and the cake to set.
Once chilled, cut into squares and serve.