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1
Scrape seeds from vanilla bean with tip of a paring knife into a small heavy saucepan. Add pod, milk, and 3 Tbsp sugar and bring just to a boil, whisking to dissolve sugar.
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2
Meanwhile, whisk together flour, cornstarch, remaining sugar, and a pinch of salt in a bowl. Whisk together whole egg and yolk in another bowl, then whisk in flour mixture until smooth.
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3
Add one third of hot milk mixture to yolk mixture (to temper) in a slow stream, whisking, then add to milk mixture remaining in saucepan, whisking. Bring custard to a boil over medium heat, whisking, then boil, whisking, 2 minutes. Remove from heat and stir in butter until melted.
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4
Transfer pastry cream to a bowl and chill, its surface covered with wax paper, until no longer hot, at least 30 minutes.
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5
Preheat oven to 400u00b0F with racks in upper and lower thirds. Line 2 large baking sheets with parchment paper.
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6
Peel and core apples, then cut into 1/8-inch-thick wedges.
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7
Roll out pastry on a lightly floured surface with a lightly floured rolling pin 1/8 inch thick and cut out 8 (6-inch) rounds using inverted bowl or plate as a guide. Punch out center of each round with cookie cutter and transfer to baking sheets.
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8
Discard vanilla pod and transfer pastry cream to pastry bag. Snip a 1/3-inch opening and pipe pastry cream onto pastry in a ring around center (there will be some cream left over). Arrange overlapping apple slices on pastry, covering the cream, then brush with melted butter.
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9
Bake tarts, switching position of sheets halfway through, until edges of apples are pale golden, about 15 minutes.
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10
Meanwhile, stir together corn syrup and water until smooth.
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11
Remove baking sheets from oven and flip each tart over with a large spatula. Generously brush pastry with some of corn syrup mixture.
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12
Continue to bake, switching position of sheets halfway through, until pastry is shiny and golden, about 15 minutes more. Cool 10 minutes.
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13
Serve tarts, pastry side up, warm or at room temperature.
