Make the crust: Preheat oven to 350 degrees. Process gingersnaps, granulated sugar, and salt in a food processor until finely ground. Add butter, and process until combined. Divide mixture among six 5-inch pie plates, pressing into bottom and up sides. Place on a baking sheet, and bake until darkened and firm, 11 to 13 minutes. Let cool 5 minutes. Using an offset spatula, carefully unmold crusts, and return to sheet to cool. (Crusts can be stored in pie plates in airtight containers for up to 2 days.)
Make the filling: Sprinkle gelatin over cold water in a bowl. Let stand until softened, about 5 minutes.
Combine pumpkin, egg yolks, 1/4 cup granulated sugar, milk, salt, and spices in a saucepan over medium heat. Cook, stirring, until mixture begins to thicken, about 8 minutes. Do not boil. Remove from heat. Stir in gelatin mixture until dissolved. Let cool completely.
Beat egg whites with a mixer until soft peaks form. Gradually add remaining 1/2 cup granulated sugar, beating until stiff peaks form. Whisk one-third of beaten egg whites into cooled pumpkin mixture. Gently fold in remaining whites. (Filling can be refrigerated, covered, overnight.) Spoon into pie shells, and refrigerate for at least 1 hour (or up to 2 hours).
Make the garnish: Using a vegetable peeler, shave pumpkin into thin, wide ribbons. Bring water, brown sugar, cinnamon sticks, and ginger to a boil in a saucepan over medium heat. Cook until reduced by half, about 5 minutes. Add pumpkin pieces, and reduce heat. Simmer gently until tender and translucent, about 8 minutes. Pour into a bowl, and let cool completely. (Garnish can stand at room temperature for up to 3 hours.) Top each pie with 3 or 4 pumpkin pieces, discarding syrup. Serve immediately.
This recipe also can make two 8-inch pies; be sure to leave the crust in the pie plate before adding the filling. If you'd like to make fresh pumpkin puree, place the halved, seeded sugar pumpkin remaining from the garnish on a rimmed baking sheet, cut sides down. Roast in a 400-degree oven until soft, about 50 minutes. Scoop flesh from skin, and puree until smooth.