Make the dough: Put the flour, sugar, and salt in a food processor and pulse a few times to blend. Add the cold butter and pulse until the pieces are the size of green peas. With the processor running, pour in half the water and blend for 15 to 30 seconds. Add the vanilla and more of the remaining water as needed so the dough starts to hold together in a ball.

Transfer the dough to a floured surface and gently knead a few times to bring it together into a smooth ball. Divide the dough in half and shape each piece into a 6-inch square. Tightly cover with plastic wrap and refrigerate for at least 1 hour.

Meanwhile, caramelize the apples: Heat a large skillet over medium heat, add the butter, and place the apple pieces in the pan, flat side down. Cook, turning the apples occasionally, until lightly caramelized and golden brown all over. Turn the heat down if the butter starts to burn. Take your time with this step so that the apples are nicely caramelized; it should take about 15 minutes. Remove the pan from the heat and set aside to cool.

Preheat the oven to 350F. Grease a 9-by-13-inch baking dish with butter.

Make the citrus filling: Put the orange and lemon in a food processor and process to a fairly smooth puree. Transfer to a medium bowl and stir in the sugar, raisins, and cinnamon. Set aside.

Shape the dough: Lightly flour a work surface. Roll out one piece of dough to a 13-by-17-inch rectangle, about ⅛ inch thick. 

Transfer the dough to the prepared baking dish and gently press it over the bottom and up the sides, leaving a bit of an overhang on all sides. Roll out the second piece of dough to a rectangle the same size as the first and set aside.

Sprinkle the breadcrumbs evenly over the dough in the baking dish. Arrange the caramelized apples over the breadcrumbs in one layer. Pour the citrus mix evenly over the apples.

Lay the second piece of dough over the filling, taking care to drape it rather than stretch it. Pinch the dough edges together to seal on all sides, trim off all but about 1½ inches, and then crimp the dough to make the edges look neat and pretty. Brush the surface of the pie with the beaten egg. With a sharp knife, cut about three 4-inch diagonal slits in the top of the dough. Sprinkle the sugar evenly on top.

Bake the pie until the crust is a deep golden brown and the filling is cooked through, poking an apple through one of the slits to see if it is tender, about 50-70 minutes. Check the pie halfway through, and if the top crust is browning too quickly, cover it loosely with aluminum foil. When the pie is done, remove it from the oven and let rest for about 20 minutes, so the filling can firm up a bit. The strudel can be stored in an airtight container in the refrigerator for 5 days.

Serve the pie warm, cut into slices.