Upsidedown Caramelized Apple Tart

Preparation
To make the pâte brisée, sift the flour onto a work surface and make a well in the center. Put egg yolk, salt, and water in the well with flavorings such as sugar. Pound the butter with a rolling pin to soften it, add it to the well, and work the ingredients in the well with the fingers of one hand until thoroughly mixed. Using a pastry scraper, gradually draw in the flour from the sides of the well and continue working with both hands until coarse crumbs form. If the crumbs seem dry, sprinkle with another tablespoon of water; the crumbs should be soft but not sticky. Press the dough gently together into a ball; it will be uneven and unblended at this point.

To blend (fraiser) the dough, sprinkle the counter with flour and put the dough on it. With the heel of your hand, push the dough away from you, flattening it against the counter. Gather it up, press it into a rough ball, and flatten it again. This flattening motion evenly blends the butter with the other ingredients without overworking the dough. Work quickly so the butter doesn't get too warm. Continue until the dough is as pliable as putty and pulls away from the counter in one piece, 1 to 2 minutes. Shape it into a ball, wrap, and chill until firm, 15 to 30 minutes.

Peel and halve the apples; scoop out the cores with a melon baller. Melt the butter in the mold, sprinkle in the sugar and cook over medium heat without stirring until it starts to brown and caramelize. Stir gently, then continue cooking until the caramel is deep golden brown, 6 to 8 minutes total. Let it cool in the pan for 3 to 5 minutes—the butter will separate but this does not matter.

Arrange the apples in the mold in concentric circles with the cut sides standing vertical—the caramel will help to anchor them. Pack them as tightly as possible as they will shrink during cooking. Cook the apples over medium heat until the juice starts to run, about 8 minutes, then raise the heat and cook them as fast as possible until the underside is caramelized to deep golden and most of the juice has evaporated, l5 to 25 minutes. With a two-pronged fork, turn the apples one by one so the upper sides are now down in the caramel. Continue cooking until this second side is brown also and almost all the juice has evaporated, l0 to 20 minutes more. The time will vary very much with the variety and ripeness of the apples, and can take up to an hour in total. Let them cool to tepid while heating the oven to 400°F/200°C/Gas 6.

Roll the pastry dough to a round just larger than the mold. Wrap the dough around the rolling pin and transfer it to cover the apples. Tuck the edges down around the apples, working quickly so their warmth does not melt the dough. Poke a hole in the center to allow steam to escape. Bake the tart until the pastry is firm and lightly browned, 20 to 25 minutes. Take the tart from the oven and let it cool for at least l0 minutes, or until tepid. Tarte Tatin can be made up to 12 hours ahead and kept in the mold in the refrigerator (if using a skillet, the tart must be turned out immediately).

To finish, if necessary, warm the tart in the mold on the stove or in the oven before you turn it out: this softens the caramel and loosens the apples. Select a flat platter with a lip to catch any juices; set the platter on top of the tart pan and flip the tart onto the platter. Be careful because you can be splashed with hot juice. Cut into wedges to serve.