Preheat the oven to 375°F and spray an 8-inch square baking pan with nonstick cooking spray. Melt the butter in a small skillet over medium heat, cooking just until the butter begins to brown.
Whisk together the corn syrup, brown sugar, pecans, eggs, vinegar, vanilla extract, salt, and brown butter in a large bowl to combine. Pour the filling into the prepared baking pan and bake for 20 to 25 minutes, until the filling is slightly thickened. Transfer the filling to a clean bowl and chill in the refrigerator for at least 1 hour.
Place the flour, sugar, and salt in the bowl of a food processor and pulse a few times to combine. Add the butter and shortening and pulse until the mixture resembles coarse crumbs. With the machine running, add the water in a slow, steady stream, blending just until the dough comes together.
Transfer the dough to a lightly floured work surface and divide in four pieces. Shape each piece of dough into a flat disk and wrap in plastic wrap. Chill the dough in the refrigerator for at least 1 hour, or overnight.
Line two baking sheets with parchment paper. Transfer one disk of dough to a lightly floured work surface and roll into a circle slightly larger than 8 inches in diameter. Trim the edges of the dough to form an 8-inch square. Cut the dough into 4 squares and arrange on the prepared baking sheets. Repeat with the remaining disks of dough, so that you have 16 squares in total.
Preheat the oven to 375°F. Place a large spoonful of the filling in the center of one square of dough. Brush the edges of the square lightly with the egg wash and place another piece of dough over the filling, pressing gently with your fingers to seal. Using the tines of a fork, crimp the edges of the pie, then cut a few small slits in the top of the dough. Brush the top lightly with egg wash and place on the baking sheet. Repeat with the rest of the pie dough and filling.
Bake for 30 to 35 minutes, until the crusts are golden brown. Remove from the oven and let cool slightly before serving.