Preheat the oven to 350 degrees F. Lightly grease 2 (10-inch) cake pans with 1 teaspoon of the butter. Line the bottoms with parchment paper, and butter the parchment with 1 teaspoon of the butter. Set aside.

Puree the bananas, sour cream, eggs, and vanilla in a food processor.

Sift the cake flour, baking soda, baking powder, and salt into the bowl of an electric mixer fitted with a paddle attachment.

In the bowl of an electric mixer, cream the remaining 10 tablespoons butter and the sugar. On medium speed, add the dry ingredients in 3 additions, alternating with the wet, scraping the sides of the bowl and beating well after each addition.

Pour the batter into the prepared pans. Bake the cake is lightly browned and bounces back when touched with your fingers, about 40 minutes. Let cool for 10 minutes before turning out onto a wire rack to cool completely. Peel off the parchment paper.

To assemble the cake, using a long serrated knife, trim off the rounded top of each cake layer to make them flat. Place the bottom layer on a wire rack set over a baking sheet. Spread the pastry cream evenly over the top. Place the next cake layer upside down on the first. Wrap the cake around the sides with double layers of plastic wrap to keep the pastry cream from oozing out. Gently press down on the top layer. Remove the plastic wrap and scrap off any excess pastry cream.

Slowly pour 1 to 1 1/2 cups of the ganache onto the top of the cake, letting it run down the sides. Let sit for 5 minutes, then, in several applications, slowly pour on the remaining ganache until the cake is completely covered. Chill until ready to serve.

Pastry Cream:

Combine 2 cups of the cream, 1/2 cup plus 1 tablespoon of the sugar, and the vanilla bean in a medium saucepan. Bring to a gentle boil over medium heat, whisking to dissolve the sugar. Remove from the heat.

Combine the remaining 2 tablespoons heavy cream, the remaining 1 tablespoon of sugar, the egg yolks, and cornstarch in a medium bowl and whisk until pale yellow. Slowly add 1/2 cup of the hot cream to the egg yolk mixture, whisking constantly until smooth. Add the egg yolk mixture back to the hot cream, and whisk well to combine. Return to medium heat, and cook, stirring constantly with a heavy wooden spoon, until the mixture thickens enough to coat the back of a spoon, about 5 minutes.

Remove from the heat and strain through a fine mesh strainer into a clean container. Cover with plastic wrap, pressing it directly against the surface to prevent a skin from forming. Refrigerate until well chilled, about 4 hours. Whisk until smooth before using.

Chocolate Ganache:

Scald the cream in a small saucepan over medium heat. Remove from the heat.

Place the chocolate in a heat-proof bowl. Add the hot cream and let sit for 2 minutes, then whisk until smooth. Let cool slightly before pouring on the cake.