In a large bowl, mix together flour, sugar, and salt. Add cold butter and mix until the mixture resembles coarse crumbs. Stir in ice water, a tablespoon at a time, until the dough comes together. Form into a disk, wrap in plastic, and refrigerate for 30 minutes.

Apple Filling Preparation

Prepare the Apple Filling

In a large bowl, combine sliced apples, brown sugar, cinnamon, lemon juice, and vanilla extract. Toss to coat and set aside.

Assemble and Bake

Assemble the Tart

Preheat the oven to 375°F (190°C). Roll out the chilled dough on a floured surface and fit it into a tart pan. Pour the apple mixture into the crust and spread evenly.

Bake the Tart

Bake for 35-40 minutes, or until the apples are tender and the crust is golden brown. Remove from the oven and let it cool slightly.

Caramel Rum Sauce Preparation

Make the Caramel Rum Sauce

In a saucepan over medium heat, melt sugar until it turns golden brown. Stir in butter until melted, then carefully add cream and rum. Stir until smooth and remove from heat.

Serving

Serve the Tart

Slice the warm apple tart and serve with scoops of vanilla ice cream drizzled with caramel rum sauce.