Pour the whole milk and heavy cream into a medium saucepan. Set the pan over medium heat and warm the mixture until it reaches a gentle simmer, stirring frequently to prevent the dairy from scorching.
In a medium mixing bowl, whisk the whole egg, egg yolks, granulated sugar, and cornstarch until completely blended and smooth.
While whisking continuously, gradually spoon a small amount of the hot milk mixture into the egg mixture. Continue adding several more spoonfuls in stages, whisking throughout, until the eggs have been gently warmed without curdling.
Pour the tempered egg mixture into the saucepan containing the remaining warm milk and cream.
Cook over medium-low heat for 5–8 minutes, whisking constantly, until the custard becomes thick enough to coat the back of a spoon. Do not allow the mixture to boil, as excessive heat may compromise its smooth texture.
Remove the saucepan from the heat. Add the butter pieces, vanilla extract, and salt, stirring until the butter has fully melted and all ingredients are evenly incorporated.
Pass the custard through a fine-mesh sieve into a clean container to remove any small lumps and create a silky consistency.
Press plastic wrap directly against the surface of the custard to prevent a skin from developing. Refrigerate for at least 1 hour, or until thoroughly chilled. Stir the custard well before assembling the dessert.
Arrange a single layer of vanilla wafers across the bottom of a 9-inch square baking dish, trifle bowl, large serving bowl, or individual dessert dishes.
Cover the wafers with a layer of sliced bananas, then spread half of the chilled custard evenly over the fruit.
Repeat the layering process with additional wafers, bananas, and the remaining custard. Complete the dessert with a final covering of whole or crushed vanilla wafers.
Cover the assembled pudding tightly with plastic wrap and refrigerate for 4–8 hours. This resting period allows the flavors to combine and the wafers to soften.
Immediately before serving, finish the banana pudding with freshly whipped cream.