Make the custard: In a medium saucepan, whisk together sugar, flour, and salt. In a separate bowl, whisk egg yolks, then gradually whisk in 1/2 cup of the milk until smooth.

Pour remaining milk into the saucepan with dry ingredients. Cook over medium heat, whisking constantly, until thickened and bubbling, about 8-10 minutes. Cook for 1 more minute.

Temper the eggs: Slowly whisk about 1 cup of the hot milk mixture into the egg yolk mixture. Then, pour the egg mixture back into the saucepan.

Cook the custard for 2-3 more minutes over medium heat, whisking constantly, until very thick. Remove from heat and whisk in butter and vanilla.

Pour custard into a bowl and press plastic wrap directly on the surface. Chill in refrigerator for at least 2-4 hours, or until completely cold.

Assemble the pudding: In a 9x13-inch dish, spread a thin layer of custard. Top with a layer of vanilla wafers, then a layer of sliced bananas, then a layer of custard. Repeat layers, ending with a final layer of custard.

Preheat oven to 350°F (175°C). For the meringue, beat room temperature egg whites and cream of tartar with a mixer on medium speed until soft peaks form.

Gradually add sugar, 1 tbsp at a time, while beating on high speed until stiff, glossy peaks form.

Spread meringue over the final custard layer, creating swirls and peaks. Bake for 8-12 minutes, or until meringue is golden brown. Cool for 30 minutes before serving.