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To make the dough: Fit the bowl of a food processor with the chopping blade and place the flour, sugar, and salt in the bowl. Pulse to combine.
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Add the butter and pulse in 2-second bursts until the butter pieces are all the size of peas or smaller. Drizzle the water over top and pulse in 1-second bursts until the dough just begins to come together in small clumps and holds together when squeezed.
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Transfer the dough to a piece of plastic wrap and use the wrap to help gather the crumbly mass into a ball, then press down gently to flatten it to about 1 1/2" thick. Chunks of butter should still be visible. Don’t work the dough any more than necessary; handling it gently at this point will help prevent a tough crust.
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Wrap the dough tightly and refrigerate it for at least 30 minutes or up to overnight.
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To prepare the apples: Use a melon baller or teaspoon cookie scoop to remove the blossom and stem end of each apple. Peel the apples, then cut them into thirds, leaving behind the core. The apple pieces will be slightly different sizes — that's OK.
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In a 12" skillet, spread the sugar across the bottom in an even layer. Cook over medium heat until the sugar begins to melt at the edges. Use a heatproof spatula to bring the melted sugar toward the center of the pan; continue to cook until the sugar is fully melted and is deep amber in color, 5 to 6 minutes. Remove the pan from the heat and whisk in the butter, vinegar, vanilla, and salt. The caramel may seize and appear clumpy at this stage; that’s OK.
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Return the pan to medium heat and add the apple pieces. Cook, stirring frequently, for 15 to 20 minutes, until the caramel has reduced enough to coat the apples and the back of the spatula, and the apples are tender enough that the tip of a sharp paring knife slides in easily. Throughout the cooking process, adjust the heat to ensure that the caramel is gently bubbling: if it’s too languid, it won’t reduce sufficiently; if it’s too vigorous, it will evaporate before the apples have reached the desired doneness.
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To assemble the apple tarte tatin: Preheat the oven to 400°F with a rack in the center and lightly grease a 9" round pan. Using tongs, place the apple pieces in the pan starting at the outer edge: Arrange them rounded-side down in concentric circles, overlapping each piece slightly. Initially, it may look as though all the apples won’t fit, but pack them into the pan as tightly as possible. (Because they’ll shrink slightly during baking, arranging them tightly will prevent gaps from forming.) Scrape any caramel remaining in the bottom of the pan on top of the apples. Set aside to cool for 15 minutes.
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Lightly flour a work surface. Remove the dough from the refrigerator and roll the dough into a 9 1/2" round. Lay the dough over the apples, tucking the edges of the dough into the pan. With a sharp knife, cut a few steam vents in the center.
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Set the pan on a rimmed baking sheet and transfer to the oven. Bake for 35 to 40 minutes, until the dough is deep golden brown.
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Remove the pan from the oven and let cool for 45 minutes, then invert a serving plate on top of the cake pan and turn the tarte tatin out onto the plate. Slice and serve warm or at room temperature.
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Storage information: Store leftover apple tarte tatin, lightly covered, at room temperature for several days. Freezing is not recommended.
