In a medium saucepan, combine the milk, sugar, flour, and salt over medium heat. Whisk continuously until the mixture thickens, about 5-7 minutes.

In a separate bowl, whisk together the eggs and egg yolks. Gradually temper the egg mixture by slowly adding 1/2 cup of the hot milk mixture while whisking constantly.

Pour the tempered egg mixture back into the saucepan with the remaining milk mixture. Cook over medium heat, whisking constantly, until the custard thickens further and coats the back of a spoon, about 3-5 minutes.

Remove the saucepan from heat and stir in the vanilla extract. Let the custard cool slightly, stirring occasionally to prevent a skin from forming.

In a 9x13-inch baking dish or deep serving dish, layer vanilla wafers at the bottom. Add a layer of sliced bananas on top, followed by a layer of custard. Repeat the layers until all ingredients are used, ending with custard on top.

Preheat your oven to 375°F (190°C).

To prepare the meringue topping, beat the egg whites and cream of tartar in a clean bowl using an electric mixer until soft peaks form. Gradually add the 1/4 cup sugar and continue beating until stiff peaks form.

Spread the meringue evenly over the pudding, ensuring it touches the edges of the dish to seal. Create decorative peaks with a spatula or spoon.

Bake the pudding in the preheated oven for 10-12 minutes, or until the meringue is golden brown.

Remove from the oven and let cool to room temperature before serving. You can also chill it in the refrigerator if you prefer it cold.