Set out an 8-by-8-inch square dish, or a 2-quart dish at least 2 inches deep. In a medium bowl, whisk the egg yolks, cornstarch, sugar and 1/2 cup of the milk together.

Heat the remaining 2 1/2 cups milk in a medium saucepan over medium-high heat until it just starts to simmer, then remove it from the heat. Whisking constantly, gradually add hot milk to the yolk mixture, 1/2 cup at a time, until the mixture is warm. Pour it back into the saucepan with the remaining milk.

Stir in the vanilla. Cook over medium heat, whisking constantly and scraping the bottom of the pan, until thickened, about 3 to 4 minutes. The pudding should coat a spoon and hold a trail drawn through it. Because it contains egg yolks, check that it reaches 160°F with a food thermometer. Remove from the heat and whisk in the butter.

Layer vanilla wafers and freshly sliced bananas in the dish, aiming for at least two layers of each. Pour the hot pudding over them and gently shake the dish to help it fill the gaps.

Let the dish cool briefly, then cover and refrigerate for 8 hours. Get it into the refrigerator within 2 hours of cooking. If you like, top with whipped cream and crumbled wafers when you are ready to serve.