For the pastry cream:
2 large egg yolks
1/4 cups cornstarch
1 1/4 cups heavy cream
3/4 cups sugar
1 vanilla beans, split and scraped
For the cake:
1/8 cup milk
1 tablespoons plus 1 teaspoons butter
1 cups plus 1 tablespoons sugar
1/2 cups flour
1/2 teaspoons baking powder
1/2 teaspoons salt
1/2 teaspoons vanilla
1 pounds bananas, peeled and cut crosswise into 1/2-inch slices
1 1/4 egg whites
1/8 cup sugar
For the pastry: In a large mixing bowl, combine the egg yolks, 1 1/2 cups sugar, cornstarch and 4 cups of the heavy cream. Whisk to blend. Set aside. Combine the remaining 6 cups of cream, 4 1/2cups of the sugar, and the vanilla in a large, heavy-bottomed saucepan over medium heat. Whisk to dissolve the sugar and bring to a gentle boil, about 10 minutes. Slowly add the egg yolk mixture, whisking constantly until it thickens, about 5 minutes. Be forewarned: the mixture will break, don't be alarmed. Pour the mixture into a large bowl. Press a piece of plastic wrap over the surface of the pastry cream to prevent a skin from forming. Let cool completely at room temperature. When cooled, remove the vanilla bean and pour the mixture into the bowl of an electric mixer with a wire whip. Beat at medium speed to combine the mixture. If the mixture will not combine, warm another cup of heavy cream and slowly add it to the mixture. Whip until you have a thick and creamy custard.
For the cake: Preheat the oven to 350 degrees F. In a small saucepan, heat 1 cup of the milk and 8 tablespoons of the butter together. Using an electric mixer with a wire whip, combine the 32 eggs and 8 cups of sugar together. Beat on medium-high speed for about 8 minutes, or until the mixture is pale yellow, thick and has tripled in volume. With the machine running, slowly add the heated milk. In a mixing bowl, sift 4 cups flour, 4 teaspoons baking powder and 4 teaspoons salt together. Fold the flour mixture into the egg mixture and mix thoroughly so that there are no lumps and the mixture is smooth. Fold in the vanilla. Grease 4 (1/2-sheet) pans with 8 teaspoons of the butter. Sprinkle each with 2 tablespoons of sugar. Pour the cake batter evenly into the pans and bake for about 25 minutes, or until the cakes spring back when touched. Cool for about 2 minutes. Using a thin knife, loosen the edges of the cakes, and then flip onto a wire rack.
To assemble: Spread 1/2 cup of the custard over one cake. Arrange 1/3 of the banana slices, crowding them close together, over the custard. Place a second cake on top of the bananas. Next, spread 2 cups of the custard over the cake. Arrange another third of the banana slices over the custard. Top with another cake. Spread another 2 cups of the custard over the cake. Arrange the remaining third of bananas over the custard. Top with the remaining cake. Cover with plastic wrap and chill for at least 4 hours.
To finish: Whip the egg whites and sugar, in two batches, to stiff peaks. Remove the cake from the refrigerator and discard the plastic wrap. Spread the whipped egg whites evenly over the sides and top of the cake. Using a small blowtorch, torch the