Combine all-purpose flour, brown sugar, and salt in a food processor. Cut in the cold unsalted butter until crumbly, then mix in vodka until a dough forms. Refrigerate for at least 2 hours.

Preheat your oven to 350°F. Roll out the chilled dough and transfer it to a pie pan, crimping the edges. Dock the surface to prevent bubbling, and bake for 35-40 minutes until golden brown. Cool completely.

In a bowl, whisk together granulated sugar, cornstarch, and a pinch of salt. Heat half-and-half until simmering, temper it into the egg mix, and cook over medium heat until thickened and boiling. Cool and refrigerate for 2 hours.

In a saucepan, boil sugar and water until it reaches 300°F. Stir in butter, then whisk in heavy whipping cream and dark rum. Cool before chilling in the refrigerator.

Spread the chilled pastry cream into the cooled crust, layer with sliced ripe bananas, and drizzle the cooled banana rum sauce over the top before serving.