Line a 9x13-inch pan with a snug layer of graham crackers, breaking pieces to fit.
Whisk pudding mix with cold milk for about 2 minutes until thick and smooth.
Fold in the whipped topping and vanilla until no streaks remain.
Spread half of the creamy filling over the cracker layer.
Add a second layer of crackers. Spread on the remaining filling.
Top with a final layer of crackers, pressing gently to level the surface.
Stir frosting with 1 tbsp milk to loosen. Warm 10–15 seconds if needed, then spread evenly over the top.
Cover and chill at least 6 hours, ideally overnight, until the crackers soften.
Slice into squares and serve cold.