Preheat oven to 425 degrees.
In a large bowl, stir together the flour, sugar and salt. Using a pastry cutter. two knives, or your fingers, cut the butter into the flour mixture until the texture resembles coarse cornmeal, with butter pieces no larger than small peas. Add the water and mix with a fork just until the dough pulls together.
Transfer the dough to a floured work surface, pat into a ball and flatten into a disk. Lift the dough and give it a quarter turn. Lightly dust the top of the dough or the rolling pin with flour as needed, then roll out into a round at least 12 inches in diameter and about ¼ inch thick. Transfer to a parchment lined baking sheet.
In a medium sized bowl, combine the apples, brown sugar, flour, cinnamon, nutmeg, allspice, and lime juice. Stir gently to combine.
Spoon the apples onto the dough, leaving 2 inches clear around the edge. Spoon the dulce de leche onto the apples, then sprinkle with the walnuts.
Wrap the edge of the dough up and over the apple mixture, then whisk together the egg and milk in a small dish and use it to wash the edge of the dough. Sprinkle with the demerara sugar, then bake for 30 minutes, or until the dough is golden brown. Allow to cool for a few minutes before serving.