Whisk 2 1/2 cups flour and salt together in a bowl. Cut in shortening with 2 knives or pastry blender until the mixture resembles coarse crumbs. Add ice water, 1 tablespoon at a time, tossing with a fork, until flour mixture is moistened and dough comes together. Do not add more water than you need. Divide the dough in half and shape into 2 balls. Wrap in plastic wrap and refrigerate, at least 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Line a 9-inch pie pan with wax paper.
Melt 1/2 cup unsalted butter in a saucepan over medium heat. Stir in 3 tablespoons flour to form a paste. Stir in 1/4 cup white sugar, 1/2 cup brown sugar, 1/4 cup water, vanilla extract, cinnamon, and nutmeg; bring to a boil. Reduce heat and simmer until filling mixture is thickened slightly, about 3 minutes.
Melt 1/3 cup butter in a separate saucepan over low heat. Add 1/2 cup brown sugar and whisk until smooth, about 2 minutes. Stir in pecans.
Spread pecan mixture evenly onto lined pie pan. Roll out one ball of dough; line pie pan over pecan mixture. Toss apples with filling mixture; pour into pie plate. Roll out second ball of dough and place over apple filling. Crimp edges of bottom and top crusts together. Cut several small slits into top crust to vent. Place pie on foil-lined baking sheet.
Bake pie in the preheated oven until crust is golden brown, about 1 hour.
Remove pie from oven and allow to cool slightly, 5 to 10 minutes. Place a serving plate gently over pie and carefully flip to invert, being careful not to come in contact with molten sugar. Peel off waxed paper; cool for 1 hour.