This layered banana dessert pairs crunchy vanilla cookies with creamy pudding. For the ultimate in banana flavor, use very ripe bananas, They're softer, sweeter and packed with flavor.
Whisk milk, sugar, cornstarch, egg yolks and salt together in a medium saucepan. Cook over medium-low heat, whisking occasionally, until bubbling, 8 to 10 minutes. Continue cooking, whisking occasionally, until thickened, about 2 more minutes. The pudding should still be pourable and will thicken as it cools. Remove from heat and stir in butter and vanilla.
Place 12 cookies in one layer in the bottom of an 8-inch-square baking dish. Top with half the banana slices, then pour in half the pudding, smoothing it evenly over the bananas. Repeat with 12 more cookies, the remaining banana and the remaining pudding. Place a sheet of plastic wrap directly on top of the pudding to prevent a skin from forming. Refrigerate for at least 4 hours or preferably overnight. Coarsely chop the remaining 3 cookies and sprinkle over the top of the pudding before serving.
To make ahead: Cover and refrigerate for up to 3 days.