For the pastry: Put the flour, sugar and salt in a large bowl. Add the butter pieces and pinch with the fingers to create a crumb texture. Make a well in the middle, and pour in the water and vanilla. Quickly work in the flour to create dough. Do not over-mix. Divide into 2 disks, wrap in plastic and refrigerate 15 minutes before rolling out.
Preheat the oven to 400 degrees F (200 degrees C).
For the filling: Peel, core and halve the apples. Set a 12-to-14-inch/30-to-36-cm tarte tatin pan on the stove and put in the butter and sugar. Let them melt together, going from slushy, to stiff, to mealy, to liquid caramel in about 20 minutes.
Lay in the apple halves upright in concentric circles. You can use a wooden spoon laid across the pan to brace them. Pack them as tightly as you can, and when no more apples fit, know that two to three more halves will eventually, because apples shrink as they cook. So, keep some extra halves at the ready to poke in when gaps appear. Cook the apples in the pan for about 20 minutes.
The apples will be caramelized on one side. Using a fork and spoon, turn them over to caramelize the other side. If they break up as you do this, don't worry because the good side will be facing up when you flip the tart out. Continue cooking, about 20 minutes more, until the apples are fully caramelized.
Remove the pan from the heat. Lay the pastry round on top and tuck it into the edges like a blanket. Now, slip the pan into the oven and bake until the crust is golden and crisp, 15 to 20 minutes. Remove from the oven and let cool about 10 minutes before turning out.
Or, wait until just before serving so the crust stays crisp. Warm the tart in its pan on a burner until it moves. Then, flip it onto a platter.