In a small, heavy-bottomed saucepan, combine the sugar and water and bring the mixture to a boil, stirring often. Cook, stirring occasionally, until the mixture is a deep caramel color and has the consistency of thin syrup, 10 to 15 minutes. Remove from the heat. Stir in the cream, return the saucepan to high heat, and boil the sauce until it regains the consistency of a thick syrup, about 2 minutes. Cool.
The sauce can be refrigerated until ready to use. Allow it to reach room temperature before drizzling it over the pie.
Combine the half-and-half and butter in a small, heavy-bottomed saucepan over medium heat. Heat the mixture until a thin paperlike skin appears on the top. Do not boil. Add the chocolate and vanilla and stir until the chocolate melts and the mixture is smooth.
Remove from the heat and let cool.
The sauce can be kept refrigerated for several days, but must be returned to room temperature before serving.
Preheat oven to 350 degrees.
To prepare custard filling, in a mixing bowl, combine the egg yolks, cornstarch, and 1 cup of the heavy cream. Whisk to blend well. Set aside. Combine the remaining 2 cups cream, the sugar, and the vanilla bean 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: this mixture must break and look curdled otherwise it will not set up properly and it will be runny. Pour it into a glass bowl. Press a piece of plastic wrap down over the surface of the mixture 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 fitted with a wire whip. Beat at medium speed to combine the mixture. If it will not combine, warm another 1/2 cup heavy cream and slowly add it to the mixture. Whip until you have a thick and creamy custard.
To prepare pie crust, in a mixing bowl, combine the graham cracker crumbs, sugar and mashed banana. Mix thoroughly. Add the butter and mix well. Press the mixture into a 9-inch pie pan. Bake until browned, about 25 minutes. Remove the pan from the oven and cool, for about 10 minutes. To assemble, spread about 1/2 cup of the custard on the bottom of the crust. Arrange about a third of the banana slices, crowding them close together, over the custard. Next, spread 1 cup of the custard over the bananas. Arrange another third of the banana slices close together over the custard. Top with 1 cup of the custard and the banana slices. Top with the remaining custard, covering the bananas completely to prevent them from turning brown. Cover with plastic wrap and chill for at least 4 hours. Cut the pie into wedges and serve with a drizzle of caramel and chocolate sauce. Top with the whipped cream and shaved chocolate. Sprinkle with powdered sugar.