Place the oven rack in the center position and preheat to 400°F (204°C).
Unfold the puff pastry sheet on a work surface covered with parchment paper. It's best to cut the tart shells out when the pastry is semi-frozen. So it's easier to unroll, cut and layer onto your baking sheet.
Line a baking sheet with parchment paper.
Using a cookie cutter, create a 4-inch bottom layer, then poke several holes into the dough to prevent the dough from rising too much as it bakes. Brush with a thin coating of egg wash.
For the top layer, cut a 4-inch round, then using a 3-inch round, cut a hole out of the center of the 4-inch round. Place on top of the base. Transfer the cut out shells to the baking sheet.
Brush the puff pastry top with egg wash. Bake the puff pastry shells until puffed and golden brown, about 20 minutes. Remove from the oven and allow to cool.
Custard Filling
Heat milk and vanilla bean over medium heat in a medium saucepan. When the milk reaches 90°F (32°C), stir in 3 ounces of sugar. Allow the milk to come to a simmer, with small bubbles forming on the edges, and then turn off the heat.
Scrape the inside of the vanilla bean with a knife and put the seeds into the milk, whisk to combine.
In a mixing bowl, combine the remaining sugar with the cornstarch.
Place the egg yolk in a separate bowl and add the cornstarch and sugar mixture. Whisk thoroughly to combine, leaving no residual cornstarch.
In the bowl containing the egg mixture, gradually whisk in the hot milk into the egg yolk mixture to temper the eggs.
Return the egg and milk mixture to the pan and cook over medium-high heat, whisking vigorously until the cream boils and is well thickened. The custard should not have a residual starch flavor. Turn off the heat and whisk the butter into the custard.
Transfer filling to a bowl and chill over an ice bath. Once cooled, cover and keep refrigerated until ready to use. If the custard solidifies after cooling, place the custard in a mixing bowl, then using the whip attachment, beat the mixture until smooth.
Honey Ginger Glaze
In a small saucepan, steep ginger in the water and honey mixture over medium-low heat, until the mixture reduces by half (⅛ cup). The glaze should be slightly thinner than the honey so that it can be easily brushed on top of the persimmons, yet sticks the fruit.
Assembly
Fill each tart shell with a generous amount of custard filling.
To prepare the persimmon slices, cut the fruit into quarters, then remove the skin from the fruit. Thinly slice the persimmons to about ⅛ to ¼-inch wedges.
Arrange the persimmons slices into a cylindrical fan on top of the custard filling. Generously glaze the slices.