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1
Put oven racks in upper and lower thirds of oven and preheat oven to 400u00b0F. Line 2 large baking sheets with parchment paper.
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2
Peel apples, reserving skins. Stir together water and 6 tablespoons sugar in a 1-quart heavy saucepan, then scrape in seeds from vanilla bean and add pod. Add reserved apple skins and bring to a boil, stirring until sugar is dissolved. Remove from heat and let stand 10 minutes. Pour through a medium-mesh sieve into a bowl, pressing hard on and then discarding solids.
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3
Roll out 1 pastry sheet into a 10-inch square on a lightly floured surface with a lightly floured rolling pin, then cut in half and transfer to 1 of baking sheets. Repeat with remaining pastry sheet, transferring to second baking sheet.
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4
Halve apples lengthwise, then core with a melon-ball cutter or a small spoon and cut crosswise into very thin slices (1/8 inch thick or less), keeping apple halves intact. Fan apple slices slightly, keeping apple shape, then arrange 3 halves in a row on each piece of pastry, leaving a 2/3-inch border on all sides.
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5
Brush border lightly with syrup and fold over about 1/3 inch to touch edges of apples. Crimp edges with a fork, then brush apples and edges of pastry with syrup (reserve remaining syrup). Brush apples completely with melted butter, then sprinkle evenly with remaining 2 tablespoons sugar.
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6
Bake tarts in upper and lower thirds of oven 20 minutes, then switch position of sheets and reduce oven temperature to 375u00b0F. Bake tarts until edges are golden, 15 to 20 minutes more. (If pastry edges aren't browning evenly, rotate tarts 180 degrees on sheets halfway through baking.)
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7
While tarts bake, boil remaining syrup over moderate heat until reduced to about 1/4 cup, 30 seconds to 1 minute.
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8
Let tarts stand 5 minutes, then brush warm apples with remaining syrup.
