Make the flaky pastry: Blend the flour with the salt in a food processor. Add the butter and process until well blended, about 8 seconds. Add the water and process just until the water has been absorbed and the mixture looks like wet sand. Transfer the pastry to a work surface and knead lightly until it comes together. Pat the dough into a disk, wrap in plastic, and refrigerate until firm, about 1 hour.

On a lightly floured surface, roll the pastry into an 11 ½-inch round. Drape the dough over a rolling pin, then open it out on a baking sheet lined with plastic wrap and refrigerate until ready to use.

Meanwhile, make the apple filling: Spread the sugar evenly in the bottom of a 9-inch cast-iron skillet. Scatter the butter over the sugar and drizzle with the vanilla. Arrange the apple halves on their sides in the skillet in 2 concentric circles with all of the apples facing the same direction. Pack the apples closely together. Place 1 apple half in the center to fill the empty space.

Set the skillet over moderately low heat and cook the apples until the surrounding syrup becomes a thick, golden brown, 45 minutes to 1 hour. Baste the apples with the syrup regularly with a bulb baster. The liquid should remain at a gentle bubble.

Preheat the oven to 425°F. Place the skillet on a baking sheet and bake the apples for 15 minutes. Remove from the oven. Drape the pastry over the apples and carefully push the edge of the pastry down inside the pan. Return the skillet to the oven and bake for 25 to 30 minutes, or until the pastry is golden and the juices are bubbling over.

Remove the tart from the oven. Immediately invert a platter with a lip over the skillet. Quickly and carefully invert the tart onto the platter so the apples are on top. If any apples stick to the skillet, set them back into the tart. Serve warm or at room temperature, with dollops of crème fraîche, whipped cream, or ice cream.