In a large mixing bowl, vigorously whisk the instant vanilla pudding with the chilled milk for about 2 minutes, or until it thickens noticeably. Once set, gently incorporate the thawed whipped topping using a folding motion until the mixture is smooth and airy.
Line the bottom of a 9-by-13-inch baking dish with a single, even layer of graham crackers. Break some pieces if needed to fill in gaps completely.
Spread half of the prepared pudding mixture over the base layer of crackers, smoothing it into an even surface with a spatula.
Lay down a second full layer of graham crackers over the pudding, ensuring complete coverage.
Distribute the remaining pudding mixture evenly over the second cracker layer and spread to smooth.
Arrange a final layer of graham crackers over the top of the pudding to complete the assembly.
To prepare the frosting, remove the lid and foil from the container and microwave for 10 to 15 seconds to slightly soften it—just until it becomes spreadable. Stir thoroughly and pour the frosting over the top graham layer, then use a spatula to spread it evenly from edge to edge.
Cover the dish tightly with plastic wrap and refrigerate for a minimum of 6 hours, or ideally overnight, to allow the crackers to absorb moisture and achieve a cake-like consistency.
When ready to serve, cut into squares and serve chilled. Keep any leftover portions refrigerated and consume within 5 days.