In a large mixing bowl, using electric hand mixer, combine milk and instant pudding mix. Mix on low speed until combined and thickened, about two minutes.
In a separate large mixing bowl, combine softened cream cheese and sweetened condensed milk. Beat together on medium-high speed until no lumps remains and mixture is smooth.
Gently fold in the Cool Whip, using a rubber spatula, just until combined.
Transfer to the bowl with the pudding and stir gently until completely combined.
Line the bottom of a 9×13-inch casserole dish with 1 bag of the cookies.
Top with sliced bananas in an even layer.
Next, top the bananas with the whipped cream mixture and smooth out even with a rubber spatula.
Lastly, lay out the second bag of Chessmen cookies.
Cover dish with plastic wrap and refrigerate at least 3 hours (overnight is even better!)
Scoop the banana pudding onto serving plates and enjoy!