Vanilla-Spiced Caramel and Pear Tart

Ingredients:
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
1/4 cup (1/2 stick) unsalted butter
1/2 vanilla bean, split lengthwise in half
2 cinnamon sticks, broken in half
2 whole star anise
6 whole cloves
1/4 cup sugar
1 large egg
Generous pinch of coarse kosher salt
1 1/2 tablespoons all purpose flour
1 egg white, beaten to blend
Ingredient Info: All-butter puff pastry is available frozen at Whole Foods markets. One 14- to 16-ounce package contains two 11-inch rounds. Whole star anise is available in the spice section of some supermarkets.
10-inch-diameter springform pan

Instructions:
Roll out pastry to 12-inch square and trim off corners, forming slightly rounded crust. (If using all-butter puff pastry, roll out to 12-inch round.) Transfer pastry to 10-inch-diameter springform pan, pressing pastry firmly onto bottom and 1 to 1 1/2 inches up sides of pan. Freeze crust until firm, 1 to 1 1/2 hours. DO AHEAD: Can be made 1 day ahead. Cover and keep frozen.
Melt butter in heavy large skillet over low heat. Add sugar and next 4 ingredients. Scrape in seeds from vanilla bean; add bean. Increase heat to medium-high and cook, stirring until sugar melts and turns brown (color of peanut butter), about 3 minutes. Reduce heat to medium; add pears, rounded side down. Cook until pears are almost tender, turning and moving skillet around occasionally to ensure even cooking, 10 to 15 minutes, depending on ripeness of pears. Carefully turn pears over; continue to cook until pears are very tender, about 10 minutes longer. Remove skillet from heat; cool pears in skillet with spices (pears will release juice while cooling).
Melt butter in small saucepan over low heat. Scrape in seeds from vanilla bean; add bean. Add cinnamon, star anise, and cloves. Increase heat to medium; cook until butter is golden (watch to avoid burning), 3 to 4 minutes. Remove vanilla bean and spices from butter; discard. Whisk sugar, egg, and coarse salt in medium bowl. Whisk in flour. Gradually whisk browned butter into egg mixture. DO AHEAD: Pears and filling can be made 6 hours ahead. Cover separately; let stand at room temperature. Whisk filling before using.
Position rack in center of oven; preheat to 400°F. Brush frozen crust with beaten egg white. Pour filling into crust; spread evenly over bottom of crust (filling will be very thin). Using slotted spatula, remove pears from skillet, allowing excess syrup to drain back into skillet; reserve syrup. Arrange pears, rounded side up, atop filling (about 9 around edge and 3 in center of pan).
Bake tart until crust is deep golden and filling is set and brown at edges, about 1 hour. Run small knife around sides of pan to loosen tart. Release pan sides. Transfer tart to platter. Let stand, uncovered, at room temperature to cool slightly until just warm.
Just before serving, boil syrup in skillet until reduced to generous 1/3 cup, 1 to 2 minutes. Discard spices. Drizzle syrup over pears. Serve tart slightly warm.