In a standing electric mixer or food processor, combine the flour, sugar, salt, and butter until the butter is in small, pea-sized pieces.

Add the ice water and mix until the dough just begins to hold together. If it looks too dry, add a sprinkle more water.

Knead the dough for a couple of seconds until it comes together, and shape it into a flat disk. Chill for 30 minutes.

Pour the strained quince poaching liquid into a tarte Tatin pan or cast iron skillet and cook over moderate heat until thick and syrupy. Remove from heat.

Arrange the poached quince quarters snugly in the pan, rounded side down.

Roll the dough into a circle larger than the pan, drape it over the quince, tucking in the edges.

Bake in a 375F (190C) oven for approximately 45 minutes, until the dough is deep golden brown.

Let the tart rest on a cooling rack for a few minutes, then overturn a serving platter or baking sheet over the tart and flip it over.

Serve warm or at room temperature.