Whisk the sugar, flour, and salt together in a medium saucepan. Gradually whisk in the whole milk until the mixture is smooth and no dry pockets remain.

Cook the milk mixture over medium heat, stirring constantly, until it thickens and just begins to bubble. Keep the pan moving so the flour does not settle or scorch.

Place the egg yolks in a bowl and slowly whisk about 1 cup of the hot milk mixture into them. Pour the tempered yolks back into the saucepan while whisking continuously.

Continue cooking for 1–2 minutes, stirring constantly, until the custard is thick and glossy. Do not allow it to boil aggressively, because high heat can cause scorching or curdled eggs.

Remove the saucepan from the heat and stir in the butter and vanilla extract until completely incorporated. The warm custard should smell creamy and fragrant.

Layer vanilla wafers, banana slices, and warm custard in a serving dish. Repeat the layers, finishing with custard on top.

For meringue, beat the egg whites with the salt until foamy, then gradually add the sugar and beat until glossy stiff peaks form. Beat in the vanilla, spread the meringue over the pudding so it touches the edges, and bake at 350°F for 12–15 minutes until lightly browned.

Cool the pudding if it has been topped with baked meringue, then cover and refrigerate for at least 2 hours. Four hours is better for a fully set custard and pleasantly softened wafers.