Preheat the oven: Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone mat.
Cream the butter and sugar: In a large bowl, beat the butter at room temperature with the brown sugar until the mixture is light and fluffy, about 2-3 minutes.
Add the egg and banana: Add the egg and mashed banana to the butter and sugar mixture. Beat until well combined.
Add the dry ingredients: In the same bowl, add the banana pudding mix, flour, and baking soda. Mix on low speed until all the dry ingredients are fully incorporated into the dough.
Fold in the white chocolate chips: Stir in 1 cup of white chocolate chips with a spatula until evenly distributed throughout the dough.
Shape the cookies: Using a cookie scoop #20 or a regular spoon, form balls of dough and place them on the prepared baking sheet, leaving space between each cookie for spreading.
Chill the dough: Place the baking sheet with the dough portions in the freezer for 5 minutes. This step helps the cookies retain their shape while baking.
Bake: Bake the cookies for 7-8 minutes, or until the edges are golden and the centers are slightly soft.
Decorate: Remove the cookies from the oven and allow them to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely. Once cooled, top with extra white chocolate chips and drizzle melted white chocolate over the top.