Lightly grease a standard 9x13 inch baking dish. Arrange a single, solid layer of graham crackers on the bottom of the prepared dish. Break or cut the crackers as needed to cover the bottom completely, without overlapping. Set the remaining crackers aside.

In a large mixing bowl, whisk together the two packages of instant vanilla pudding mix and the 3 cups of cold milk for about two minutes, until the mixture starts to thicken substantially. Let the pudding mixture sit undisturbed for 5 minutes to fully set.

In a separate medium bowl, combine the 1 cup of creamy peanut butter and the 1/2 cup of powdered sugar. Stir well until the mixture is smooth and slightly malleable. Add the peanut butter mixture to the set vanilla pudding. Stir gently until the mixture is mostly combined but still slightly streaky.

Gently fold in the entire container of thawed whipped topping until the filling is uniform, light, and creamy. Do not overmix, or the volume will deflate.

Pour half (approximately 3 cups) of the peanut butter filling over the graham cracker base layer. Use a rubber spatula to gently spread the filling into an even layer, ensuring it reaches all the corners.

Arrange a second layer of graham crackers directly on top of the peanut butter filling, again making sure there are no gaps or overlaps. Pour the remaining half of the peanut butter filling on top of the second cracker layer and spread it smoothly. Place a third and final layer of graham crackers directly on top of the remaining filling.

Prepare the Chocolate Glaze: In a small saucepan over low heat, combine the 1 cup of chocolate chips and the 1/4 cup of butter. Stir constantly until the chocolate and butter are fully melted and smooth. Remove from heat and stir in the vanilla extract and the 3 tablespoons of cold milk.

Gradually whisk in the 1 cup of sifted powdered sugar until the glaze is smooth, shiny, and lump-free. If the glaze is too thick, add milk 1 teaspoon at a time until it reaches a pourable consistency.

Pour the warm chocolate glaze evenly over the top layer of graham crackers, spreading gently to ensure the entire top surface is covered. Cover the dish tightly with plastic wrap (ensure the wrap does not touch the glaze). Refrigerate the cake for a minimum of 8 hours, or preferably overnight.

Slice and serve the chilled cake. The extended chilling time allows the graham crackers to fully soften, creating the classic cake-like texture.