Peel and halve the apples; scoop out the cores with a melon baller or the point of a paring knife.
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 a deep golden brown, 6 to 8 minutes total. Let 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 vertically. 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 a deep gold and most of the juice has evaporated, 15 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, 10 to 20 minutes more. The time will vary with the variety and ripeness of the apples, and can take up to 1 hour total. Let them cool to tepid while heating the oven to 400°F (200°C).
Roll out 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 the apples' warmth does not melt the dough. Poke a hole in the center of the dough 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 10 minutes, or until it is tepid. Tarte tatin may be made up to 12 hours ahead and kept in the mold in the refrigerator (if using a cast-iron skillet, the tart must be turned out immediately).
To finish, if necessary, warm the tart in the mold on the stove top before you turn it out, to soften the caramel and loosen the apples. Select a flat platter with a lip to catch any juices; invert 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.