Preheat oven to 350°F and line a baking sheet with parchment paper.
In a large bowl, beat together butter and sugars until smooth.
Add pudding mix, mashed banana, egg, and vanilla. Mix well.
In a separate bowl, whisk together flour, baking soda, and salt.
Slowly add dry ingredients to the wet mixture, mixing between additions.
Fold in crushed Nilla wafers and white chocolate chips.
Scoop dough using a 1.5-inch cookie scoop and place on baking sheet.
Bake for 10–12 minutes. Tops will look slightly underdone, but bottoms should be golden brown.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack.