Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Mash bananas in a small bowl until smooth. Set aside.
Crush vanilla wafers in a ziplock bag with a rolling pin, leaving some texture.
Whisk together flour, baking soda, and salt in a medium bowl. Set aside.
Cream butter and both sugars until light and fluffy, about 2-3 minutes.
Add eggs one at a time, beating well after each addition.
Mix in vanilla extract and dry banana pudding mix powder.
Stir in mashed bananas until combined (mixture may look slightly curdled).
Gradually add flour mixture, mixing on low just until combined.
Fold in crushed vanilla wafers, white chocolate chips, and nuts (if using).
Chill dough for at least 30 minutes.
Scoop 2 tablespoon portions onto prepared baking sheets, spacing 2 inches apart.
Press extra vanilla wafer pieces and white chocolate chips on top of each cookie.
Bake for 10-12 minutes, until edges are golden but centers still look slightly underdone.
Cool on baking sheets for 5 minutes before transferring to a wire rack.