Process flour, salt, and sugar together in a food processor until combined, about 3 seconds. Add butter and pulse until butter is the size of large peas, about ten 1-second pulses.
Using a fork, mix sour cream and 1/3 cup ice water in a small bowl until combined. Add half of the sour cream mixture to flour mixture; pulse for three 1-second pulses. Repeat with the remaining sour cream mixture. Pinch dough with fingers; if the dough is floury, dry, and does not hold together, add 1 to 2 tablespoons ice water and process until dough forms large clumps and no dry flour remains about three to five 1-second pulses.
Turn dough out onto the work surface. Divide dough into 2 balls and flatten each into 4-inch disk; wrap each disk in plastic and refrigerate until firm but not hard, 1 to 2 hours, before rolling. (Dough can be refrigerated for up to 24 hours. Let thoroughly chilled dough stand at room temperature for 15 minutes before rolling.)
Mix 1/2 cup granulated sugar, brown sugar, salt, zest, and apple pie spice in a large bowl; add apples and toss to combine. Transfer apples to Dutch oven (do not wash bowl) and cook, covered, over medium heat, stirring frequently, until apples are tender when poked with fork but still hold their shape, 15 to 20 minutes. (Apples and juices should gently simmer during cooking.) Transfer apples and juices to the rimmed baking sheet and cool to room temperature, about 30 minutes. While apples cool, adjust oven rack to lowest position, place the empty rimmed baking sheet on rack, and heat oven to 425 degrees.
Remove 1 disk of dough from refrigerator and roll out between 2 large sheets of parchment paper or plastic wrap to 12-inch circle, about 1/8 inch thick. (If dough becomes soft and/or sticky, return to refrigerator until firm.) Remove parchment from one side of the dough and flip onto a 9-inch pie plate; peel off the second layer of parchment. Working around circumference, ease dough into plate by gently lifting the edge of dough with one hand while pressing into plate bottom with another hand. Leave dough that overhangs plate in place; refrigerate until dough is firm, about 30 minutes.
Meanwhile, roll the second disk of dough between 2 large sheets of parchment paper or plastic wrap to 12-inch circle, about 1/8 inch thick. Refrigerate, leaving dough between parchment sheets, until firm, about 30 minutes.
Set a large colander over the now-empty bowl; transfer cooled apples to a colander. Shake the colander to drain off as much juice as possible (cooked apples should measure about 8 cups); discard juice. Transfer apples to dough-lined pie plate; sprinkle with lemon juice.
Remove parchment from one side of the remaining dough and flip the dough onto apples; peel off the second piece of parchment. Pinch edges of the top and bottom dough rounds firmly together. Trim and seal edges of the dough and then cut four 2-inch slits on top of the dough. Brush the surface with beaten egg white and sprinkle evenly with a remaining teaspoon sugar.
Set pie on the preheated baking sheet; bake until crust is dark golden brown, 45 to 55 minutes. Transfer pie to wire rack and cool at least 1 1/2 hours. Cut into wedges and serve with ice cream (optional).