I start by whisking the flour, instant vanilla pudding mix, kosher salt, baking soda, and baking powder together in a medium bowl. Getting this step out of the way first means I'm not scrambling to measure things later while my butter mixture is sitting out. I also crush my Nilla wafers at this stage, sealing them in a zip-top bag and lightly rolling over them until I have small, uneven pieces.
Next, I beat the room-temperature butter with the brown sugar and granulated sugar until the mixture turns pale and fluffy. This usually takes less than a minute on medium speed. I always scrape down the sides of the bowl at least once so nothing gets left behind unmixed at the bottom.
I mash my banana with a fork until it's completely smooth, then add it to the butter mixture along with the egg yolk and vanilla extract. I beat this on medium speed until everything is combined. Don't panic if the mixture looks a little curdled at this point, that's completely normal here.
Once the wet ingredients look well blended, I add the flour mixture in on the lowest mixer speed. I stop as soon as the flour disappears into the dough, since overbeating at this stage is what leads to tougher, denser cookies. A few dry streaks left in the bowl are fine.
I switch to a flexible spatula for this part and fold in the crushed Nilla wafers and white chocolate chips by hand. I like doing this manually so I can control how evenly everything gets distributed without breaking the wafer pieces down too much.
I scoop rounded portions of dough, about two tablespoons each, onto my parchment-lined baking sheets, leaving a couple of inches between each one. They go into a 350°F oven, and I rotate the pans halfway through baking so everything browns evenly, front to back and top to bottom.
Once the edges turn light golden, I pull the cookies out and let them rest on the baking sheet for about five minutes before moving them to a wire rack. This short rest lets them firm up so they don't fall apart when I transfer them. I repeat the scooping and baking process with the remaining chilled dough, reusing the same parchment once it's cooled.