Make crust: Prepare dough as recipe or package label directs; chill for 1 hour. Preheat oven to 400°F; line a large, rimmed baking sheet with foil. Make filling: Toss apples with 2 Tbsp. sugar, lemon juice, tapioca and vanilla.
Make caramel: Mix 1 1/2 cups sugar and 1/4 cup water in a large, heavy saucepan. Cook over low heat, stirring, until sugar dissolves. Increase heat to high and bring to a boil. Cook, without stirring, until mixture turns dark brown, 6 to 7 minutes. Standing back to avoid splattering, pour in cream. Reduce heat slightly and stir until caramel is smooth and thick, about 2 minutes.
Roll out each dough disk into a 12-inch circle. Transfer 1 to a 9-inch pie plate. Spoon in half of apple mixture. Drizzle with about 1/4 cup caramel. Mound remaining apples on top and drizzle with another 1/4 cup caramel. Cover and chill remaining caramel (rewarm over low heat before serving with pie).
Place second dough disk over filling, leaving a 1-inch overhang. Fold top edges over and crimp to seal. Cut steam vents in top. Beat egg with 1 tsp. water and lightly brush mixture over top crust. Sprinkle with remaining 2 Tbsp. sugar. Place pie on baking sheet and bake until filling bubbles and crust is golden brown, about 1 hour. Let cool before serving.