Step 1

Preheat the oven to 375°. Line a baking sheet with parchment paper. On a lightly floured surface, roll out the puff pastry to a rough 10-by-15-inch rectangle, then trim it to a neat 9-by-14-inch rectangle. Using a ruler, cut a 3/4-inch-wide strip of dough from each side; you will have 2 long and 2 short strips. Transfer the rectangle to the prepared baking sheet and brush with the egg wash. Set the strips on each side to form a border, pressing firmly to help the dough adhere; brush the strips with the egg wash. Freeze the tart shell until chilled, about 10 minutes.

Step 2

Prick the bottom of the tart shell all over with a fork. Bake in the lower third of the oven for about 40 minutes, until the shell is puffed and golden. Press down the center of the shell slightly. Increase the oven temperature to 425°.

Step 3

Rub the bananas with the lemon juice. In a skillet, cook the sugar over moderate heat, stirring until melted. Cook without stirring until a medium-amber caramel forms, about 5 minutes. Remove from the heat; whisk in the butter, vanilla seeds and water. Add the bananas and gently turn to coat with the caramel. Arrange the bananas in the tart shell, cut sides up, leaving most of the caramel in the skillet. Drizzle 1/4 cup of the caramel over the bananas. Bake the tart for about 20 minutes, until the bananas are slightly tender. Let cool slightly, then serve with whipped cream or ice cream, passing the remaining caramel on the side.