Set the oven rack in the middle and heat the oven to 325 degrees F. If you are making the meringue, chill the mixer bowl and beaters while you prepare the custard.

In a large, heavy-bottomed saucepan, whisk the flour, 1 1/2 cups sugar, and salt until well combined.

Separate the eggs, reserving the 3 whites for the meringue. Lightly beat the yolks in a large liquid measuring cup, then whisk in the milk. Pour the egg mixture into the dry ingredients and cook over medium-low heat, stirring constantly, until the custard is thick and smooth. Remove from the heat and stir in 1 teaspoon vanilla.

Evenly cover the bottom of a 2-quart ovenproof glass dish or baking dish with one-third of the vanilla wafers.

Peel and slice the bananas, then arrange one-third of the slices over the wafers.

Spoon one-third of the warm custard over the bananas and wafers.

Repeat the layers two more times with the remaining wafers, bananas, and custard, ending with a final layer of wafers on top.

Put the reserved egg whites in the chilled bowl and beat with an electric mixer on high speed until foamy.

Add the cream of tartar, then gradually beat in the remaining 1/4 cup sugar one tablespoon at a time. Continue beating until stiff, glossy peaks form and the sugar is fully dissolved, then beat in the remaining 1 teaspoon vanilla.

Pile the meringue over the banana pudding and spread it gently to the edges of the dish.

Bake for 20 to 25 minutes, until the meringue is lightly browned. Let the pudding stand for a few minutes before serving.