Preheat your oven to 425°F. On a lightly floured surface, roll the puff pastry sheet just slightly larger than your skillet (8–9 inches is perfect) and cut it into a circle that will fit snugly over the pears. Set aside.
In a small oven-safe skillet (cast iron works best), Add butter, brown sugar, granulated sugar, vanilla, cinnamon, star anise.
Nestle the pear halves cut-side down into the caramel mixture, arranging them in a circle so they fit snugly. Lay the puff pastry circle over the pears, tucking the edges in slightly around the fruit.
Let this cook on the stove on medium heat until the caramel starts to bubble and turn brown, spilling over the puff pastry just a little bit.
Transfer the skillet to the oven and bake for 25 minutes, or until the puff pastry is golden brown, puffed, and the caramel is bubbling around the edges.
Let the tart sit for about 10 minutes so the caramel sets slightly. Then place a plate larger than the skillet on top, use oven mitts to hold both together, and flip carefully so the tart ends up on the plate, pears on top.
Slice into wedges and serve warm with a generous scoop of vanilla ice cream.