Pour the flour, salt, and sugar in the bowl of a food processor fitted with a metal blade. Pulse for a few seconds to combine.

Add the butter and pulse about 10 times, until the butter becomes the size of small peas. With the motor running, pour the ice water down the feed tube and pulse just until the dough starts to come together. Take the dough out onto a floured board and knead quickly into a ball. Wrap the dough in plastic and refrigerate for at least 1 hour.

Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.

Sprinkle some flour and roll out the dough into a flat rectangular shape slightly larger than 10 by 14-inches. With a small knife, trim the edges to size. Place the dough on the prepared sheet pan and refrigerate while you prepare the apples.

Peel the apple, cut into quarters and core the apple. Slice the apples into 1/4-inch thick slices.

Place overlapping slices of apples diagonally down the middle of the tart and continue making diagonal rows on both sides of the first row until the pastry is covered with apple slices.

Sprinkle the full 1/2 cup of sugar and dot with the butter. Bake until the pastry is browned and the edges of the apples start to brown, about 45 minutes to 1 hour. Rotate the pan once during cooking. If the pastry puffs up in one area, cut a little slit with a knife to let the air out. The apple juices will burn in the pan but the tart will be fine.

When the tart is done, heat the apricot jelly together with the rum in a small pot. Run the mixture through a sieve and brush the apples and the pastry completely with this mixture.

Loosen the tart with a spatula so it doesn't stick to the paper. Allow to cool and serve warm or at room temperature.