Arrange one package of Chessmen cookies evenly across the base of a 13x9x2-inch dish, then layer the banana slices over the cookies. In a mixing bowl, combine the pudding mix with the milk and beat thoroughly using a handheld electric mixer until well blended. In a separate bowl, beat together the cream cheese and sweetened condensed milk until the mixture is smooth and creamy. Gently fold the whipped topping into the cream cheese blend. Combine this with the prepared pudding mixture and stir until uniformly incorporated. Pour the resulting mixture over the banana and cookie layers in the dish. Top with the remaining bag of cookies. Cover and refrigerate until chilled and ready to serve.