Preheat the oven to 350 degrees F and line a large cookie sheet with parchment paper.
In a medium-sized bowl, whisk together the flour, salt, baking soda, and baking powder.
In a large mixing bowl cream together the butter and sugars until light and fluffy.
Add the dry pudding mix, eggs, and vanilla extract, and continue mixing until well combined.
Add the dry ingredients to the wet ingredients and stir until just combined, making sure not to over mix the dough.
Fold in the white chocolate chips and ¾ cup of the Nilla pieces. The dough will feel quite thick (this is the correct consistency).
Scoop even mounds of the cookie dough onto the parchment-lined cookie sheets, making sure to space them out by 1 inch.
Gently press the remaining Nilla pieces into each cookie.
Bake them for 10 minutes or until the edges start to turn golden brown and the cookies are still soft in the center.
Remove the cookie sheet from the oven and place it on a wire rack to cool for 5 minutes before enjoying.