Cut each roll in half and coat in the butter.
Place half of the remaining crushed cookies into your dry pudding mix. Stir it up. Then dredge each buttery dinner roll piece into the pudding and cookie mixture.
Repeat process with the other half of the rolls, and top with the remaining cookie crumbs and white chocolate chips. Spray a piece of plastic wrap and cover the pan. Allow to rise until double in size or it reaches the top of the pan.
Bake in a preheated 350˚F oven for 25 to 35 minutes until the bread is browned.
For the cream cheese icing: In a medium-large bowl, beat the cream cheese, butter, powdered sugar, and vanilla with an electric hand mixer. Add milk one Tablespoon at a time until desired consistency. Put icing in a plastic bag, cut off the corner and decorate the top of the warm monkey bread.