For the Caramelized Pecans:

Line a jelly-roll pan with aluminum foil; smooth out any wrinkles. Coat lightly with flavorless vegetable oil using a pastry brush.

1) Put the sugar and water in a very small saucepan and stir to combine. Bring to a boil over medium heat and cook until it turns light amber in color. Immediately remove from the heat. Quickly drop the pecan halves into the caramel one at a time and use two forks to help them submerge and become coated with caramel. Use the forks to remove the nuts from the caramel, allow the excess to drip off, and place them right side up on the prepared pan. Repeat with the remaining pecan halves. If the caramel thickens, reheat gently until fluid. The caramel will continue to darken while you are dipping, which is okay. Allow the nuts to cool completely and harden at room temperature. Store for up to 3 days in an airtight container.

For the Crust:

Position a rack in the middle of the oven. Preheat the oven to 350°F. Coat a 10-inch loose-bottomed fluted tart pan with nonstick spray; set aside.

1) Put the nuts and sugar in the bowl of a food processor fitted with the metal blade attachment and process until very finely ground.

2) Pulse in the melted butter until well blended. Pat into the prepared pan, creating an even layer across the bottom and up the sides.

3) Bake for 8 to 9 minutes. The crust might look puffy; use the back of a soup spoon to press the crust down and back into shape along the pan’s sides and bottom. Bake for 8 to 9 minutes more, or until very lightly colored. Cool the tart pan on a rack. Store at room temperature for up to 1 day, loosely covered with aluminum foil.

For the Filling:

Prepare the filling and topping the day of serving. At least 4 hours before serving, stir the sugar and water together in a nonstick skillet. Bring to a boil over medium-high heat and cook until it turns a light to medium golden amber. Meanwhile, peel the bananas. Remove the skillet from the heat and immediately slice the bananas into ½-inch rounds right into the skillet. Toss well with the caramel. Immediately scrape into the prepared crust and set aside. Prepare the pastry cream as directed in the recipe. Scrape the warm (not hot) pastry cream over the bananas and spread until smooth using a small offset spatula. Let the pastry cream cool slightly at room temperature, and then refrigerate for at least 6 hours or until firm enough to slice. After it chills, you may cover it with a piece of plastic wrap pressed directly onto the surface.

For the Topping:

Up to 2 hours before serving, peel and cut the banana in half and slice lengthwise into thin slices. Place these along the edge of the tart, using the curve of the fruit to follow the shape of the tart. There should be one ring of banana slices going around the entire tart. Beat the cream, sugar, and vanilla in the bowl of a stand mixer on medium-high speed with the wire whip attachment until firm peaks form. Pile the whipped cream attractively on top of the tart. The tart may be served immediately or refrigerated for up to 2 hours. Sprinkle the caramelized pecans on top right before serving.