Soak Raisins: Mix raisins and rum in a small bowl. Place the bowl in hot water and soak for 15-20 minutes. Drain excess rum.
Prepare Filling: Toss apples with lemon juice. Add raisins, brown sugar, cinnamon, and nutmeg. Mix well.
Preheat Oven: Set oven to 400°F (200°C). Prepare parchment paper and roll out puff pastry on it.
Assemble Strudel: Sprinkle breadcrumbs on pastry. Add apple mixture. Cut dough into strips and fold alternately over filling. Moisten edges with water.
Bake: Brush with egg wash and bake for 35-40 minutes until golden brown.
Glaze: Whisk powdered sugar, milk, cinnamon, and vanilla. Drizzle over cooled strudel.