In the bowl of a stand mixer fitted with the paddle attachment, combine flour, 1/2 cup sugar, vanilla extract, egg, and the cold butter. Knead until the dough is smooth, cover and refrigerate for 30 minutes.
Peel the apples, and cut them into chunks. In a bowl, combine apples, 1/4 cup sugar, lemon juice, cinnamon, and 1-2 tbsp rum (optional).
Preheat the oven to 350F (175C) and grease a 9-inch springform pan (26 cm).
Use 3/4 of the dough for the bottom. Roll out the dough into a circle bigger than your baking pan, transfer to your pan and bring up the dough on the sites. Stab the bottom with a fork and sprinkle breadcrumbs on top.
Roll out the rest of the dough to make the upper crust.
Add the apple chunks to the baking dish and cover with the upper crust.
Bake for 50 minutes until lightly browned. Serve still warm with vanilla ice cream or whipped cream.
The cake tastes best on the same day.