In a medium saucepan, whisk together the sugar, flour, and salt; gradually whisk in the milk until well combined and smooth.

Cook the mixture over medium heat, stirring constantly, until it thickens and comes to a boil, which should take about 5-7 minutes. Once boiling, remove the pan from the heat.

Temper the egg yolks by slowly whisking a small amount of the hot custard into them, then gradually add the tempered yolks back into the remaining custard in the saucepan, stirring continuously.

Return the saucepan to low heat and cook for 1-2 minutes, stirring constantly, until the custard thickens further, then remove it from the heat.

Stir in the butter and vanilla extract until they are fully incorporated and the butter has melted.

Arrange a single layer of vanilla wafers in the bottom of a 2-quart baking dish, followed by a layer of sliced bananas.

Pour half of the warm custard over the bananas and wafers, then repeat with another layer of wafers, bananas, and the remaining custard.

If using meringue, prepare it at this point; otherwise, cover the dish and refrigerate for at least 4 hours, or preferably overnight, for the flavors to meld.

In a clean, dry bowl, beat the egg whites with an electric mixer until foamy, then add the cream of tartar.

Gradually add the granulated sugar, beating until stiff, glossy peaks form.

Spread the meringue over the warm banana pudding, ensuring it touches the edges of the dish to prevent shrinking.

Bake in a preheated oven at 350°F (175°C) for 10-15 minutes, or until the meringue is golden brown, then let it cool before serving.