Spread one package of the Chessman cookies evenly in a 9x13-inch baking dish, allowing slight overlap.
Layer the sliced bananas uniformly on top of the cookies.
Using a large bowl, whisk the softened cream cheese with the sweetened condensed milk until velvety.
Incorporate the dry instant banana pudding mix into the bowl and whisk until fully blended.
Stir in the vanilla extract and cold whole milk until the mixture is smooth, using a spatula to scrape the bowl's sides as necessary.
Carefully fold in the thawed Cool Whip until no streaks remain, then distribute the mixture uniformly over the bananas.
Scatter the remaining package of Chessman cookies over the pudding layer.
Refrigerate the covered baking dish for no less than 4 hours.
Take out from the refrigerator, cut into portions, and serve.
