Preheat the oven to 400 degrees F and line a baking sheet with parchment paper.
On a lightly floured work surface, roll the puff pastry sheet to a thickness of between 1/4 and 1/8-inch. Using a plate or other round template as your guide, cut the pastry to a 12-inch circle. Refrigerate while you prepare the toppings.
Combine the currants and port in a small nonreactive bowl and set aside to allow the currants to plump for at least 20 minutes.
In a food processor, or using a hand mixer, combine the goat cheese with the confectioners' sugar and the vanilla and mix until smooth. Drain the currants from the port and in a small saucepan reduce the port to about 1/4 cup. Reserve the port reduction for serving.
Place the puff pastry circle onto the lined baking sheet and spread the goat cheese mixture over the puff pastry, leaving about a 1-inch border around the edges. Sprinkle the currants over the goat cheese; arrange the pear slices evenly over the goat cheese, overlapping the slices to form a fanned out pattern. Combine the apricot jam with a small amount of water; in a small saucepan or in the microwave, melt the jam. Brush the tart with the apricot jam and sprinkle with the dark brown sugar.
Bake the tart until pastry is golden brown on the bottom and sides, usually 25 to 30 minutes. Remove the tart from the oven and transfer to a wire cooling rack to cool. Serve the tart at room temperature, drizzled with some of the port reduction.