In a large mixing bowl, whisk together the instant vanilla pudding mix and cold milk until smooth. Allow it to sit for about 5 minutes until it thickens.
Once the pudding is thickened, gently fold in the creamy peanut butter until fully combined. Then, carefully fold in the thawed Cool Whip to create a light, fluffy filling.
In your 9x13-inch baking dish, arrange a layer of graham cracker squares at the bottom. This will be the base of your cake.
Spread half of the pudding mixture over the graham crackers, followed by a layer of sliced bananas.
Add another layer of graham crackers on top of the bananas, followed by the remaining pudding mixture. Top this with another layer of sliced bananas.
Top the final layer of bananas with another layer of graham crackers, covering the surface of the cake.
Melt the unsweetened chocolate in a microwave-safe bowl or using a double boiler. Once melted, mix in the softened butter, powdered sugar, vanilla extract, and milk until smooth. If the mixture is too thick, add more milk a tablespoon at a time until you reach a pourable consistency.
Pour the chocolate mixture over the top layer of graham crackers, spreading it evenly to cover all the crackers.
Cover the baking dish with plastic wrap and refrigerate the Chunky Monkey clair Cake for at least 4 hours, preferably overnight. This will allow the graham crackers to soften and the flavors to meld.