Prepare the filling: In a large mixing bowl, whisk together the instant vanilla pudding mix and 3 cups cold milk until well combined and beginning to thicken, about 2 minutes. Let stand for 5 minutes to firm up slightly.
Fold in the whipped topping: Gently fold the thawed whipped topping into the pudding mixture until fully incorporated and no streaks remain. Be careful not to overmix as this will deflate the cream.
Assemble the first layer: Line the bottom of a 9×13-inch baking dish with a single layer of graham crackers, breaking them as needed to create a complete layer that covers the entire surface.
Add cream layer: Spread approximately one-third of the pudding mixture evenly over the graham crackers, making sure to reach all the way to the edges.
Continue layering: Create a second layer of graham crackers over the pudding layer, then add another third of the pudding mixture. Repeat with a final layer of graham crackers and the remaining pudding mixture.
Top with crackers: Finish with a final layer of graham crackers on top. This will be covered with the chocolate glaze later.
Prepare the chocolate glaze: In a medium saucepan over medium heat, combine butter, cocoa powder, 1/4 cup milk, and granulated sugar. Bring to a gentle boil, stirring constantly, and cook for 1 minute.
Finish the glaze: Remove from heat and stir in vanilla extract. Gradually whisk in powdered sugar until the glaze is smooth and glossy with no lumps.
Add the topping: While the chocolate glaze is still warm and pourable, spread it evenly over the top layer of graham crackers, working quickly before it sets.
Chill thoroughly: Cover the dish with plastic wrap and refrigerate for at least 8 hours or overnight. This resting time is crucial as it allows the graham crackers to soften into cake-like layers.
Serve chilled: Cut into squares and serve directly from the refrigerator for the best texture and flavor.