Chop the toasted pecans by hand or the best texture; a food processor will grind them too small. The filling should be added to the pie shell right after it comes out of the oven. Therefore, start step 5 about 20 minutes after the pie shell goes in the oven.
FOR THE CRUST: Mix flour, sugar, and salt in food processor fitted with steel blade. Scatter butter and shortening over dry ingredients and pulse until mixture resembles coarse cornmeal, 10 to 15 seconds. Turn mixture into medium bowl.
Sprinkle 3 tablespoons ice water over mixture. With blade of rubber spatula, use folding motion to mix. Press down on dough with broad side of spatula until dough sticks together. If dough does not come together, stir in remaining ice water, 1 teaspoon at a time, until it does. Shape dough into ball with hands, then flatten into 4-inch disk. Dust dough lightly with flour, wrap tightly in plastic wrap, and place in refrigerator for at least 1 hour or up to 2 days.
Roll dough on lightly floured surface into 13-inch circle and transfer to 9-inch pie pan, preferably glass. Press dough into corners and sides of pan, being careful not to stretch dough. Trim edges of dough to make 1/2-inch overhang. Tuck overhanging dough under so that folded edge is flush with rim of pan. Flute edge, (see illustration 1).
Chill shell until firm, about 1 hour. Prick sides and bottom with fork and line entire shell with heavy-duty aluminum foil, pressing foil firmly against shell and extending it over fluted rim (illustration 2). Prick foil with fork (illustration 3) and return shell to refrigerator while oven is heating.
Adjust oven rack to center position and heat oven to 400 degrees. Bake, pressing once or twice with mitt-protected hands, if necessary, to flatten any puffing, until crust is firmly set, about 15 minutes. Remove foil and continue to bake until bottom begins to color, about 10 minutes longer. Remove from oven, and set aside while preparing the filling.
Lower oven temperature to 275 degrees. Place pie shell in oven if not still warm.
FOR THE FILLING: Melt butter in medium heatproof bowl set in skillet of water maintained at just below simmer. Remove bowl from skillet; mix in sugar and salt with wooden spoon until butter is absorbed. Beat in eggs, then corn syrup and vanilla. Return bowl to hot water; stir until mixture is shiny and warm to the touch, about 130 degrees. Remove from heat; stir in pecans.
Pour mixture into warm shell; bake until center feels set yet soft, like gelatin, when gently pressed, 50 to 60 minutes. Transfer pie to rack; let cool completely, at least 4 hours. Serve pie at room temperature or warm, with lightly sweetened whipped cream or vanilla ice cream.
TO MAKE AHEAD: The pie can be stored at room temperature, wrapped tightly in foil, for up to 2 days. If you prefer to serve the pie warm, cool the pie thoroughly so that it sets completely, then rewarm it in a 250-degree oven for about 15 minutes.