In a large mixing bowl, cream together the butter, brown sugar, and white sugar using an electric mixer until well mixed and fluffy.
Add two eggs, an extra egg yolk, and vanilla to the bowl and mix until combined.
Pour in the banana instant pudding mix and mix until fully incorporated.
In a separate bowl, combine the flour, baking soda, and salt and mix to combine.
Gradually add the dry mixture to the wet mixture and mix until a dough forms.
Fold in the white chocolate chips and mix until evenly distributed.
Chill the dough for 30 minutes in the refrigerator.
Preheat the oven to 350 degrees Fahrenheit and line a baking sheet with parchment paper.
Using a medium cookie scoop, portion the dough into two tablespoon-sized balls and place them on the prepared cookie sheet, leaving two inches between each.
Top each cookie dough ball with several extra white chocolate chips.
Bake for 10-12 minutes until the edges are set and lightly browned.
Let cookies cool for 2-3 minutes on the pan before transferring to a wire rack to cool completely.