Make the dough: Pulse the flour, sugar and salt in a food processor until combined. Add the shortening and pulse until it disappears, about 30 seconds. Add the butter; pulse until the mixture looks like coarse meal with pea-size bits of butter. Beat the egg with 2 tablespoons ice water in a bowl, then add to the processor and pulse once or twice (stop before the dough gathers into a ball). Turn the dough out onto a sheet of plastic wrap and pat into a disk. Wrap tightly and chill until firm, at least 1 hour.
Roll the dough into a 12-inch round on a floured surface. Ease into a 9-inch glass pie plate (not deep dish), then fold the overhanging dough under itself and crimp the edges with your fingers. Cover tightly with plastic and chill 30 minutes.
Position a rack in the lower third of the oven; put a baking sheet on the rack and preheat to 450 degrees F. Make the filling: Scrape the vanilla seeds into a skillet; add the butter and the vanilla pod. Cook over medium-high heat until the butter browns, about 5 minutes. Remove from the heat.
Whisk the eggs, corn syrup, sugar, vinegar and salt in a bowl. Slowly scrape in the butter mixture, whisking; discard the vanilla pod. (If using vanilla extract instead, add it now.)
Spread the chopped pecans in the crust and pour in the corn syrup mixture. Top with the remaining 3/4 cup pecan halves. Place the pie on the hot baking sheet, reduce the oven temperature to 350 degrees F and bake until the crust is golden, 20 to 30 minutes. Remove from the oven and cover the entire top with foil, then continue baking until set, 25 to 30 minutes. Discard the foil. Cool completely on a rack.