Line 2 baking sheets with parchment. Coarsely grate 2 sticks (1 cup) of the butter on the large holes of a box grater onto one of the prepared baking sheets. Toss the butter with a sprinkle of flour and arrange the butter in an even layer on the prepared sheet. Freeze until rock solid, about 30 minutes.
Whisk together the flour and 1/2 teaspoon salt in a large bowl. Fold in the frozen butter pieces. Slowly drizzle in 1/2 cup ice water, whisking with a fork until small pieces form and the dough just starts to come together, being careful not to overwork it. Turn the dough out onto a floured surface and roll out into a 12-by-1/4-inch-thick circle. Transfer to the other prepared baking sheet and refrigerate until ready to use.
Preheat the oven to 375 degrees F.
Meanwhile, peel, core and halve the apples. Cut all but one of the apple halves in half again. Put the lemon juice in a large bowl and toss the apple pieces with the lemon juice; set aside. Cut the remaining 1 stick (8 tablespoons) butter into small cubes; set aside.
Sprinkle the sugar over the bottom of a large ovenproof skillet and cook over medium heat, stirring frequently, until melted and amber brown, about 6 minutes. Remove the skillet from the heat and whisk in the cubed butter until melted and smooth (the sugar will start to bubble as the butter cools it down).
Place the apple half in the center of the skillet rounded-side down. Arrange the remaining apple quarters rounded-side down in a tight circle around the center apple, making sure there is as little space between the apples as possible (any space left between apples will cause the tart to collapse when flipped out of the skillet).
Place the pastry round on top and carefully tuck the edges down around the apples. Poke the pastry all over with a fork. Bake until the apples are cooked through and the pastry is golden brown, about 1 hour. Let cool for 5 minutes in the skillet; any longer will cause the caramel to harden and make it very hard to get the tart out of the skillet. Place a serving dish or cake stand upside down on top of the skillet. Using oven mitts, carefully flip the skillet over to invert the tart onto the serving dish. Slice and serve with vanilla ice cream.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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