In a large bowl, whisk the instant pudding mix with the cold milk for roughly 2 minutes, or until the texture is smooth and the mixture begins to thicken. Set aside to rest.
In a separate mixing bowl, beat the softened cream cheese using an electric mixer until it reaches a fluffy consistency.
Add the sweetened condensed milk to the cream cheese, continuing to mix until the two are fully blended and smooth.
Gently incorporate the previously prepared pudding into the cream mixture, folding until the mixture is uniform.
Carefully fold in about two-thirds of the whipped topping until the texture is creamy and well-mixed. Set aside the remaining one-third for the final layer.
Using a 9x13-inch baking dish or a large trifle bowl, begin assembling the dessert. Lay a single, even layer of vanilla cookies across the base.
Arrange half of the sliced bananas over the cookies in an even spread.
Spoon half of the creamy pudding mixture over the banana layer and smooth it out with a spatula.
Repeat the layering process with another set of cookies, banana rounds, and the remaining pudding mixture.
Finish by spreading the reserved whipped topping smoothly across the top layer.
Optionally, garnish the top with extra banana slices or crushed vanilla cookies for added texture and presentation.
Cover the dish and refrigerate for a minimum of 2 hours, preferably overnight, to allow the flavors to meld and the cookies to soften into the pudding.