Make the Crust: Prepare the pie crust dough and refrigerate for at least 1 hour.
Adjust the oven rack to the center position and heat the oven to 400 degrees F. Remove the dough from the refrigerator and roll it out on a lightly floured work surface to a 12-inch circle about 1/8 inch thick. Roll dough loosely around rolling pin and unroll into pie plate, leaving at least 1-inch overhang on each side. Working around circumference, ease dough into plate by gently lifting edge of dough with one hand while pressing into plate bottom with other hand. Refrigerate 15 minutes.
Trim overhang to ½ inch beyond lip of pie plate. Fold overhang under itself; folded edge should be flush with edge of pie plate. using thumb and forefinger, flute edge of dough. Use a fork to prick the dough all over the bottom and sides of the pie plate. Refrigerate dough-lined plate until firm, about 15 minutes.
Blind Bake Pie Crust: Remove pie pan from refrigerator, line crust with aluminum foil or parchment paper, and fill with pie weights or dried beans, filling the whole way up to the rim of the pie plate. Bake for 15 minutes. Remove foil/parchment and weights, rotate plate, and bake 5 to 15 additional minutes until crust is golden brown and crisp. Remove pie plate from oven.
Prepare Pie Filling: Meanwhile, melt the butter in a medium heatproof bowl set in a skillet of water maintained at just below a simmer. Remove the bowl from the skillet; stir in the sugar and salt with a wooden spoon until the butter is absorbed. Beat in the eggs, then the corn syrup and vanilla. Return the bowl to hot water; stir until the mixture is shiny and hot to the touch, about 130 degrees on an instant-read thermometer. Remove from the heat; stir in the pecans.
Bake the Pie: As soon as the pie shell comes out of the oven, decrease the oven temperature to 275 degrees F. Pour the pecan mixture into the hot pie shell.
Bake on the middle rack until the pie looks set and yet soft, like gelatin, when gently pressed with the back of a spoon, 50 to 60 minutes. Transfer the pie to a rack; cool completely, at least 4 hours or overnight.