This dessert has all the flavors of banana pudding in a handheld treat. The cake is infused with vanilla cookie crumbs and studded with bits of banana, while a silky soft mixture of white chocolate and cream create a filling that tastes just like vanilla pudding. A swoop of frosting finishes off the dessert: perfect for when a craving for banana pudding strikes.
Make the cupcakes: Preheat the oven to 350˚ F. Line 12 muffin cups with paper liners. Whisk the flour, ground cookies, baking powder and salt in a medium bowl.
Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, scraping down the bowl occasionally, 3 to 5 minutes. Beat in the eggs, one at a time. Combine the milk and vanilla in a liquid measuring cup. With the mixer on low speed, beat the flour mixture into the batter in two batches, alternating with the milk mixture in two batches. Add the chopped banana and stir with a rubber spatula until blended.
Divide the batter evenly among the muffin cups, filling each about three-quarters of the way. Bake until the cupcakes are golden and spring back when gently pressed, 25 to 30 minutes. Transfer to a rack and let cool 5 minutes in the pan, then remove to the rack to cool completely.
Make the filling: Put the white chocolate chips in a small bowl. Heat the heavy cream in the microwave until steaming, then pour over the white chocolate and let stand until softened, 3 to 5 minutes; stir until smooth. Let cool about 15 minutes, then refrigerate until thickened slightly, 5 to 15 minutes, stirring occasionally. The filling should fall off a spoon in a slow, thick ribbon.
Make the frosting: Beat the butter and salt in a large bowl with a mixer on medium-high speed until creamy, 1 to 2 minutes. Reduce the speed to low and beat in the confectioners' sugar, then increase the speed to medium high and beat until fluffy, 2 to 3 minutes. Add the vanilla and 1 tablespoon milk and beat until smooth, about 1 more minute. Beat in the remaining 1 tablespoon milk if the frosting is too stiff.
Cut a 1-inch-wide slit in the top of each cupcake, cutting about halfway down. Transfer the filling to a piping bag fitted with a small round tip. Gently squeeze each cupcake to open and pipe the filling into the cupcakes. Spread or pipe the frosting on the cupcakes; top each with a cookie and a banana slice.