1 sheet frozen puff pastry (half of 17.3-ounce package), thawed, or one 14- to 16-ounce package all-butter puff pastry (1 to 2 sheets, depending on brand), thawed
3 tablespoons unsalted butter
1/2 cup sugar
1 cinnamon stick
1 whole star anise
3 whole cloves
Generous pinch of coarse kosher salt
1/2 vanilla bean, split lengthwise
6 medium firm but ripe Anjou pears, peeled, halved lengthwise, cored
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