Preheat your oven to 350°F (175°C).
In a large mixing bowl, cream together the softened unsalted butter, brown sugar, and granulated sugar until light and fluffy (about 2-3 minutes).
Add the eggs and vanilla extract to the creamed mixture and mix until well combined.
Gradually add the banana cream instant pudding mix to the wet ingredients, mixing until fully incorporated.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add this dry mixture to the wet ingredients, mixing just until combined.
Gently fold in the white chocolate chips until evenly distributed throughout the dough.
Using a cookie scoop, drop rounded tablespoons of dough onto a baking sheet lined with parchment paper, spacing them about 2 inches apart.
Bake for 10-12 minutes, or until the edges are golden brown and the centers look slightly underbaked.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a cooling rack to cool completely.