Preheat the oven to 350°F.

Separate the eggs, putting the yolks in a medium bowl and setting the whites aside. Lightly beat the yolks until blended.

In a 3-quart saucepan over medium-low heat, combine the evaporated milk, 2 cups of the sugar, the flour, and salt and cook, stirring constantly, until the mixture is bubbling around the edges and thickly coats the back of the spoon, 25 to 30 minutes. Whisk about ¼ cup or so of the evaporated milk mixture into the beaten egg yolks to temper them, then slowly pour the tempered eggs back into the saucepan while whisking constantly. Continue whisking until fully incorporated, 2 to 3 minutes. Remove the pan from the heat and stir in the butter and vanilla extract until the butter melts. Transfer the pudding to a heatproof bowl and cover with plastic wrap, pressing directly against the surface of the pudding to prevent a skin from forming.

Place a layer of vanilla wafers along the bottom and up the sides of a 2½-quart ovenproof glass bowl or baking dish. Next, peel and slice the bananas and arrange one-third of the slices in a layer on top of the wafers on the bottom of the container. Then either spoon or pour one-third of the pudding on top of the bananas. Arrange half of the remaining vanilla wafers on top of the pudding layer, then top the pudding layer with half of the remaining banana slices, followed by half of the remaining pudding. Repeat the layers one more time with the remaining vanilla wafers, banana slices, and pudding. Set aside in the refrigerator.

In a stand mixer fitted with the whisk attachment, or in a large bowl using a handheld mixer, on medium speed, beat together the egg whites and cream of tartar until frothy. Increase the speed to medium-high and beat until soft peaks form. Then slowly add the remaining ¼ cup sugar, a little at a time, and beat until you have a very stiff meringue.

Spread the meringue on top of the pudding, taking care to cover it completely. The meringue can sit above the rim of the bowl. If you still have some banana slices, you can garnish the meringue with them.

Bake the pudding until the meringue is just golden brown, about 10 minutes. Remove from the oven and let cool for 15 minutes. Serve the pudding warm, or let cool completely, cover, refrigerate, and serve chilled.

Leftover pudding will keep, tightly covered, in the refrigerator for up to 5 days.