In a medium saucepan, whisk together 1 cup granulated sugar, flour, and kosher salt. Gradually add 2 ½ cups of milk, whisking until smooth. Cook over medium heat, stirring constantly until the mixture thickens and starts to bubble.
In a separate bowl, whisk the egg yolks. Slowly add a few spoonfuls of the hot custard mixture to the yolks to temper them, then whisk in the yolk mixture back into the saucepan. Continue cooking for 2-3 minutes until the custard is thick enough to coat the back of a spoon. Remove from heat and stir in the butter and vanilla extract.
In your 9x13-inch baking dish, layer half of the vanilla wafers at the bottom. Follow with half of the sliced bananas, then pour half of the custard over the top. Repeat the layers with the remaining wafers, bananas, and custard.
In a clean mixing bowl, beat the egg whites and cream of tartar until soft peaks form. Gradually add the remaining 1 cup of granulated sugar, continuing to beat until stiff peaks form. Spread the meringue over the custard, making sure to seal the edges to prevent shrinking.
Preheat your oven to 350°F (175°C). Bake the pudding for about 10-12 minutes or until the meringue is golden brown.
Remove the pudding from the oven and allow it to cool for at least 30 minutes before serving. This allows the layers to set and meld together beautifully.
