Special equipment: A rolling pin, a tart pan with removable bottom, a mandoline or vegetable slicer, a pastry brush A food processor, a dough scraper
Preheat the oven to 375 degrees F.
Dust a clean surface lightly with flour and roll out the round of chilled dough -- giving the dough a quarter turn every few strokes and occasionally lifting it to dust underneath with additional flour -- to a roughly 11-inch circle. Roll the dough over the pin and transfer to the tart pan, pressing the dough into the bottom and up the sides. With your fingers, pinch away the excess dough at the top of the tart pan. Transfer to the refrigerator to chill at least 30 minutes and up to overnight. If you are using a frozen pie shell, skip this step and prepare apples.
While the dough chills, make the filling. Peel, halve, and core 6 of the apples. With a mandoline, vegetable slicer, or sharp knife, cut the apples into thin 1/4-inch thick slices lengthwise. Melt 4 tablespoons of the butter in a large skillet over medium-high heat and add the apple slices. Toss to coat in butter, then add the sugar, vanilla, and salt, and continue to cook until the apples are soft and almost falling apart and most of the liquid has evaporated, about 15 to 25 minutes. Allow to cool while the dough finishes chilling.
Meanwhile, peel, halve, and core the remaining 4 apples and, with a mandoline, vegetable slicer, or sharp knife, cut into very thin 1/8-inch thick slices lengthwise. Squeeze the lemon juice over the slices to prevent browning.
When the dough has finished chilling, remove the pan from the refrigerator and spread the cooled filling over the bottom of the tart. Starting at the outside edge of the tart, make a ring of tightly overlapping 1/8-inch thick apple slices on top of the filling, tucking the last slice under the first when finished to make an unbroken circle. Make a second ring in the same manner, inside and slightly overlapping the first ring. There will be a tiny opening left in the middle: Arrange some apple slices to cover. Melt the remaining 2 tablespoons of butter and brush all over the top of the apples, being careful not to disturb the rings. Sprinkle with the remaining sugar and place on the bottom rack of the oven. Cook until the crust and the tops of the apples are golden, about 50 minutes to 1 hour and 10 minutes. Remove from the oven and cool on a baking rack. Serve warm or at room temperature.
Dissolve the salt and sugar in the ice water, then mix in the egg yolk with a fork until well combined. Place the flour in the bowl of a food processor. Sprinkle over the chilled butter cubes and then pulse on and off, until the mixture is coarse with some pea-sized lumps of butter remaining. With the processor running, add the ice water mixture in a slow stream, then pulse on and off until the dough starts to come together.
Empty the dough onto a lightly floured board and gather into a ball. Break off a small chunk at a time and smear across the board with the heel of your hand, then scrape it up with the dough scraper and set aside. Regather together all the smeared bits and form into a ball. Flatten into a 1/2-inch thick disk and wrap in plastic wrap. Place in the refrigerator to chill at least 1 hour and up to overnight.
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