Combine 2 cups of the cream, the milk, 1/2 cup of the sugar, the vanilla bean, and the vanilla seeds in a large heavy-bottomed saucepan
over medium heat. Bring to a gentle boil, whisking to dissolve the sugar. Remove from the heat.
Combine the egg yolks, eggs
, cornstarch, and 1 cup of the sugar in a medium bowl, and whisk pale yellow in color. Set aside.
Whisk 1 cup of the hot cream mixture into the egg yolks. Gradually add the egg mixture to the hot cream, whisking constantly. Bring to a simmer
, stirring constantly with a large wooden spoon to cook out the cornstarch
and the mixture thickens, about 5 minutes. (The mixture may separate slightly. If so, remove from the heat and beat with an electric mixer
until thick and smooth.) Strain through a fine mesh strainer
into a clean bowl. Cover with plastic wrap
, pressing down against the surface to prevent a skin from forming. Chill in the refrigerator for about 4 hours.
To assemble, spread 1/2 cup of the custard
over the bottom of the prepared crust, smoothing with the back of a large spoon or rubber spatula
. Arrange enough banana slices (not quite one-third) in a tight, tiled pattern over the custard, pressing down with your hands to pack them firmly. Repeat to build a second layer, using 3/4 cup of the custard and enough bananas to cover, smoothing down the layer evenly. For the third layer, spread 3/4 cup of custard over the bananas
and top with the remaining bananas, starting 1-inch from the outer edge and working toward the center. Spread 1 cup of custard evenly over the bananas to prevent discoloration. Cover with plastic wrap and chill for at least 4 hours or overnight.
In a medium bowl, whip the cream until soft peaks form. Add the remaining 2 teaspoons of sugar and the vanilla extract and whip until stiff peaks form.
Remove the pie from the refrigerator. With a sharp knife dipped in hot water, cut the pie into 10 equal slices. Transfer the slices to dessert plates. Fill a pastry bag
with the whipped cream and pipe onto each slice. (Alternately, spread the whipped cream
evenly over the pie before cutting.)
each slice with the caramel sauce and chocolate sauce, sprinkle with the chocolate shavings and confectioners' sugar, and serve.