Preheat the oven to 350°F and grease a 2-quart baking dish.

Whisk 3/4 cup of the divided sugar with the flour and salt in a medium saucepan, then gradually whisk in the milk until smooth.

Cook over medium heat, whisking constantly, for 6–8 minutes, until the mixture thickens and just begins to bubble.

Whisk the egg yolks in a heatproof bowl, slowly whisk in about 1 cup of the hot milk mixture, then whisk the yolk mixture back into the saucepan.

Cook over medium-low heat, whisking constantly, until the custard returns to a gentle boil and reaches 160°F, then boil for 1 minute.

Remove the saucepan from the heat and stir in the butter and vanilla until the butter melts.

Arrange about one-third of the wafers in the prepared dish, top with half the banana slices and one-third of the custard, then repeat with another third of the wafers, the remaining bananas and another third of the custard; finish with the remaining wafers and custard.

In a clean, dry bowl, beat the reserved egg whites with the cream of tartar at medium speed for 1–2 minutes, until soft peaks form.

Gradually beat in the remaining 1/3 cup sugar, then beat at medium-high speed for 2–3 minutes, until the meringue forms stiff, glossy peaks.

Spread the meringue over the pudding, sealing it to the edges of the dish, and bake for 15–20 minutes, until golden and the center of the meringue registers 160°F.

Rest for 20 minutes before serving, and refrigerate leftovers within 2 hours.