On a lightly floured work surface, roll the puff pastry sheet to a thickness of 1/8-inch. Using a sharp paring knife, cut a circle of pastry slightly larger than the diameter of a small nonstick skillet. (An 8-inch skillet is ideal, cutting the pastry approximately 1-inch larger.) Refrigerate while you prepare the bananas.
Preheat the oven to 400 degrees F.
Combine the sugar and water in a small (8-inch) nonstick skillet over high heat and cook, swirling from time to time, until the sugar is melted and is amber-colored, 3 to 4 minutes. Add the butter and swirl to combine. Add the sliced bananas in a decorative fashion and reduce the heat to low. Cook, swirling the pan occasionally, until the edges of the bananas look as if they're beginning to soften, 4 to 5 minutes. Remove from the heat and carefully place the puff pastry circle over the top of the pan, easing the edges so that the puff pastry sits directly on top of the hot bananas. Using a sharp knife, make several small vent holes in the pastry to allow steam to escape while baking. Bake until pastry is golden brown and flaky, 8 to 10 minutes.
Remove from the oven and twist the pan to loosen the bananas. Allow tart to sit for 1 minute before carefully inverting pan onto a serving plate. (Be very careful - the caramel is very hot!) If any of the bananas have stuck to the pan, pry them loose using a spatula and gently replace them onto the top of the tart. Serve warm or room temperature, with a scoop of vanilla ice cream.