In the bowl of a stand mixer, whip chilled cream, sugar and vanilla together until soft peaks form. Measure out 3 cups of whipped cream and refrigerate remaining whipped cream to serve as topping. In a small bowl, whisk together pudding mix and two cups of milk. Rinse the bowl of the stand mixer and using paddle attachment, beat together softened cream cheese and sweetened condensed milk until smooth.
Add pudding to stand mixer and continue to stir until combined. Then, gently fold in 3 cups of homemade whipped cream.
To assemble banana pudding:
Line bottom of 9x13 casserole dish with 20 Chessmen cookie.
Slice bananas and arrange over cookies.
Spread pudding mixture over bananas.
Finally arrange another layer of 20 Chessmen cookies on top of pudding mixture.
Cover and refrigerate for 2-3 hours or until chilled through.
Serve with additional whipped cream if desired.