Make pastry as directed, pulsing in 2 Tbsp. sugar with flour and salt. Form dough into 2 disks. Wrap in plastic; chill for at least 1 hour.
Preheat oven to 400ºF; place rack in bottom third of oven. Line a baking sheet with foil. Make filling: In a bowl, toss together apples, raspberries, lemon zest and juice, tapioca, vanilla, cinnamon and 1/2 cup sugar.
Roll out each dough disk into 12-inch circle. Transfer 1 to a 9-inch pie plate. Trim edges. Spoon in filling. Place second disk over filling. Trim edges, leaving a 1-inch overhang. Fold top edges under; crimp to seal. Cut vents in top. Lightly brush crust with beaten egg; sprinkle with 2 Tbsp. sugar. Place on baking sheet; bake until filling bubbles and crust is golden, about 1 hour. Let cool before serving.