This elegant tart has all the hallmarks of the classic banana pudding — sliced fresh bananas, creamy custard, vanilla wafer cookies and sweet meringue — with one new twist: brûléed bananas. The burned sugar over the top of the tart adds a caramel flavor that pairs well with the vanilla custard and bananas.
Make the crust: Preheat the oven to 350˚ F. Place the cookies and sugar in a food processor and process until finely ground. Add the melted butter and process until it looks like wet sand. Transfer to a 9 1/2-inch fluted tart pan and press into the bottom and up the sides. Bake until golden brown, about 12 minutes. Transfer to a rack and let cool completely.
Make the filling: Set a large fine-mesh sieve over a medium bowl. Combine the milk, heavy cream, 1/4 cup sugar, the egg yolks, cornstarch and salt in a medium saucepan; whisk until smooth. Place the saucepan over medium heat and cook, whisking constantly, until the mixture bubbles, about 5 minutes. Reduce the heat to low and continue cooking for 1 minute, whisking. Remove from the heat and immediately transfer to the sieve; press through with a rubber spatula. Stir in the butter and vanilla. Let cool to room temperature, stirring occasionally.
Peel and thinly slice 2 bananas and arrange evenly on the bottom of the crust. Gently spread the filling on top of the bananas and smooth the surface. Press a piece of plastic wrap directly on the surface; refrigerate until set, about 2 hours.
Peel and thinly slice the remaining bananas and arrange them on the tart, overlapping the slices. Sprinkle with sugar and brown with a kitchen torch.
Make the topping: Bring a few inches of water to a simmer in a medium saucepan over medium heat. Whisk the egg whites, remaining 1/2 cup sugar and the cream of tartar in a large heatproof bowl. Set the bowl over the pan; cook, whisking, until the sugar dissolves, the egg whites are frothy and the mixture is warm, 3 to 5 minutes. Remove the bowl from the pan and beat with a mixer on medium-high speed until the meringue is cool and stiff peaks form, about 5 minutes. Transfer to a piping bag; pipe the meringue around the edge of the tart. Brown with the torch.