Place the defrosted puff pastry on a lightly floured surface and carefully unroll or unfold.
Roll the dough into a 10 by 15-inch rectangle about 1/8-inch thick, dusting the dough with flour if needed to prevent sticking.
Use a large biscuit cutter or the rim of a small bowl to cut out six 5-inch rounds. Prick the rounds all over with the tines of a fork.
Arrange the dough rounds on a rimmed baking sheet lined with parchment paper, and chill until firm, 30 minutes.
Place the sugar in a small bowl. Split the vanilla bean in half lengthwise, scrape out the seeds, and rub the seeds into the sugar until evenly distributed, reserving the pod. Whisk the cornstarch and salt into the sugar, then sift the sugar through a medium mesh strainer to remove any large bits of vanilla pod.
Place the vanilla pod in a small saucepan with the butter. Cook over medium heat until the butter is golden with some dark flecks at the bottom and smells nutty and delicious, 3-5 minutes, lowering the heat as needed to prevent the butter from burning.
Pour the butter through a strainer and into a heat-proof bowl to stop the cooking and remove the vanilla pod.
Peel the pears, cut them in half lengthwise, and use a melon baller or paring knife to cut out the core. Placing the cut-side down and keeping the top of the pears together, slice each pear half lengthwise into ¼-inch pieces.
Drizzle the prepared pears with the lemon juice to prevent browning.
Position a rack in the upper third of the oven and preheat to 400ºF.
To assemble the tartlets, divide the chopped hazelnuts over the dough rounds, leaving a 1-inch border on each side.
Sprinkle the hazelnuts with half of the sugar mixture.
Carefully place a pear half on top of each dough round, fanning out the slices slightly.
Drizzle the browned butter over the pears, and sprinkle with the remaining sugar mixture.
Bake the tartlets until the pears are bubbling and the puff pastry is golden and flaky, 20-30 minutes.
Remove the tartlets to a rack to cool slightly.
While the tartlets bake, make the vanilla bean crème fraiche by combining the crème fraiche, heavy cream, vanilla paste, and sugar in a large bowl (or the bowl of a stand mixer fitted with the whip attachment). Whip the cream until it holds soft peaks. Refrigerate until needed.
Serve the tartlets warm or at room temperature topped with a dollop of crème fraiche and a sprinkle of hazelnuts.
The tartlets are best the day of baking, but leftover tarts can be refrigerated airtight for up to 3 days. Reheat in a 400ºF oven for 5-10 minutes before enjoying.
