Prepare the Pudding Mixture: In a large mixing bowl, whisk together the instant vanilla pudding mix and cold milk until smooth and thickened. Gently fold in the thawed whipped topping until fully combined.
Layer the Graham Crackers: In a 9x13-inch baking dish, arrange a single layer of graham crackers, breaking them as needed to fit.
Add the Pudding Mixture: Spread half of the pudding mixture evenly over the graham crackers. Repeat with another layer of graham crackers, then spread the remaining pudding mixture on top. Finish with a final layer of graham crackers.
Make the Chocolate Frosting: In a medium bowl, whisk together cocoa powder and powdered sugar. Stir in melted butter, milk, and vanilla extract until smooth and glossy.
Top and Chill: Pour the frosting over the top graham cracker layer, spreading it evenly. Cover the dish and refrigerate for at least 4 hours, preferably overnight, to allow the graham crackers to soften and the flavors to meld.
Serve and Enjoy: Slice into squares and serve chilled.