Place a baking sheet in oven and preheat to 400ºF. Bake pie crust until golden brown, 12 to 15 minutes. Place crust on a wire rack to cool completely.
In a small bowl, sprinkle gelatin over 1 Tbsp. cold water and set aside. In a small bowl, stir together pumpkin, pumpkin-pie spice and vanilla until well blended. Place 2 inches water in a double boiler and bring to a simmer. Whisk together egg yolks and sugar in top of double boiler and heat, whisking, until mixture registers 160ºF on an instant-read thermometer. Remove from heat and continue to whisk until cool and increased in volume, about 5 minutes. Heat gelatin mixture at 5-second intervals in microwave until dissolved, stirring between intervals. Let cool slightly, then whisk into egg yolk mixture.
Fold pumpkin mixture into yolk mixture. Fold in whipped cream. Scrape filling into cooled pie crust and spread evenly. Cover lightly with plastic wrap and refrigerate until set, at least 4 hours and up to 1 day.