How to Make It
In a small bowl, mix the egg yolk with the water. In a food processor, pulse the flour with the sugar, salt and cinnamon. Add the butter and pulse until the mixture resembles small peas. Sprinkle on the egg yolk mixture and pulse just until the pastry comes together. Turn the pastry out onto a sheet of plastic wrap and pat it into a disk. Wrap and chill until firm, at least 1 hour or overnight.
Preheat the oven to 375°. In a medium saucepan, boil the cider with the dessert wine over moderately high heat until reduced to 1 1/4 cups, about 40 minutes. Remove from the heat. Add the brown sugar and corn syrup and season lightly with a pinch of salt and pepper.
In a large bowl, toss the apples with the cornstarch; pour the hot cider syrup over and toss. Using a slotted spoon, transfer the apples into eight 1-cup ramekins. Pour the cider syrup from the bowl over the apples and top each mound with 1 piece of butter. Set the ramekins on a baking sheet and bake for about 45 minutes, stirring the apples occasionally, until browned in spots and just tender. Cover the ramekins with foil and bake for 10 to 15 minutes longer, until the apples are very tender.
Meanwhile, line a rimmed baking sheet with parchment paper. On a lightly floured work surface, roll out the pastry 1/4 inch thick. Using a 4 1/2-inch round cutter, stamp out 8 rounds; if necessary, press the scraps together to stamp out more rounds. Transfer the rounds to the prepared baking sheet and prick them several times with a fork. Crimp the edges decoratively and chill until firm, about 10 minutes.
Bake the pastry rounds for about 15 minutes, or until golden and crisp. Let cool on the baking sheet. Top each ramekin with a pastry round and bake for 5 minutes longer. Transfer the ramekins to plates and serve.
Vanilla ice cream.