Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
For the cookies: Using a stand mixer, cream the butter and sugar on medium speed until fluffy, about 2 minutes. Add the egg and vanilla and mix until smooth, about 2 minutes. In a separate bowl, combine the flour, baking powder and salt. Add the dry mixture to the butter mixture. Reduce the mixer speed to low and mix until just combined, about 30 seconds. At this point if the dough feels warm, refrigerate it until chilled, about 15 minutes. When the dough feels cool to the touch, scoop out teaspoon-size balls and drop them on the prepared baking sheets. The dough should yield 4 dozen balls. Bake until golden, 12 to 15 minutes. Let cool.
For the pudding: While the cookies are baking, warm 3 1/2 cups of the milk, granulated sugar and vanilla in a medium saucepan over medium heat until the sugar is dissolved, about 5 minutes. In a medium bowl, whisk together the egg yolks, salt and remaining 1/2 cup milk. Then whisk in the cornstarch. Slowly pour the milk mixture into the egg-cornstarch mixture, whisking constantly. When fully combined, add the mixture back to the saucepan and bring to a simmer over medium heat, stirring constantly. Cook, stirring often, until thickened to a pudding-like consistency, 10 to 15 minutes. Remove from the heat and evenly spread in a 13-by-9-inch baking dish. Cover with plastic wrap, pressing it directly on the pudding to prevent a skin from forming, and chill in the freezer for 30 minutes.
While the pastry cream is chilling, combine the cream and confectioners' sugar in the stand mixer. Whip on medium speed until thick and fluffy, 5 to 8 minutes. Fold the pudding into the whipped cream, adding 1/3 cup at a time.
To assemble the pudding: Reserve 3 cookies for the topping. Line the 9-by-13-inch baking dish with a layer of cookies, add a layer of bananas and half the pudding mixture; repeat. You should have 2 layers. Crush the reserved cookies in a resealable bag and sprinkle on top. Cover with plastic wrap and refrigerate until very cold, 2 to 3 hours.
Don't skimp on the chilling time! Homemade vanilla wafers are more durable than store bought and need time to soften. I use this recipe as a gauge for my patience and discipline. If I can wait until the cookies are soft, then I am a master of my banana pudding urges. If the cookies are still firm, the banana pudding has beaten me once again.