Preparation

• In a large (preferably 12-cup) food processor, combine the flour, salt and sugar. Add the butter and pulse in 1-second bursts until the mixture resembles coarse meal. Drizzle the ice water over the dough and pulse in 1-second bursts until it just comes together. Turn the dough out onto a work surface, gather up any crumbs and pat into 2 disks. Wrap the disks in plastic and refrigerate until chilled, about 1 hour.
  
  On a floured work surface, roll out 1 disk of the dough to a 16-inch square, a scant 1/4 inch thick. Using a small pot lid or plate (5 1/2 to 6 inches) as a template, cut out as many dough rounds as possible. Repeat with the second disk of dough. Reroll the scraps, chilling if necessary, for a total of 12 dough rounds. Transfer the pastry rounds to parchment paper–lined baking sheets and refrigerate until chilled, about 20 minutes.
  
  Preheat the oven to 375°. Spread the pecans on a baking sheet and toast for about 8 minutes, until browned and fragrant. Let cool, then coarsely chop the nuts.
  In a medium saucepan, combine 1 cup of the sugar with 1/2 cup of water. Cook over moderate heat, swirling the pan gently once the caramel begins to form, until the caramel is medium-amber, about 9 minutes. Remove from the heat and whisk in the butter, cream, 1 teaspoon of the vanilla extract and a pinch of salt. Transfer 1 cup of the caramel to a large bowl and let cool slightly. (Save any remaining caramel for later use.) Whisk the remaining 1 cup of sugar and 1 teaspoon of vanilla extract and the corn syrup into the caramel in the bowl. Whisk in the 4 unbeaten eggs until smooth. Add the pecans and a generous pinch of salt.
  Spray a 9-by-13-inch baking pan with vegetable oil spray. Pour the pecan filling into the pan and bake for about 30 minutes, until set. Let cool slightly, then scrape the pecan filling into a bowl and gently stir to recombine.
  Refrigerate until chilled, about 1 hour.
  
  Working with one baking sheet at a time, brush the pastry surfaces lightly with some of the beaten egg. Scoop a scant 1/4 cup of the pecan filling onto each round and fold the pastry over to make a turnover. Press the edges to seal and crimp with the tines of a floured fork. Refrigerate the hand pies while you prepare the rest.
  Brush the tops of the pies with the beaten egg and sprinkle with the turbinado sugar. Using a sharp knife, make a small slit in the top of each pie. Bake the pies in the middle and lower thirds of the oven for 40 minutes, until the pastry is golden brown; shift the pans halfway through baking. Let the pies cool on the baking sheets before serving.
  MAKE AHEAD
  The pies can be refrigerated for up to 4 days. Rewarm before serving.