For the pastry, place the flour, sugar, and salt in the bowl of a food processor fitted with the steel blade. Pulse a few times to combine. Add the butter and pulse 10 to 12 times, until the butter is the size of peas. With the motor running, pour the ice water down the feed tube and pulse just until the dough starts to come together. Dump onto a floured board and knead quickly into a 5-inch-round flat disk. Wrap in plastic and refrigerate for exactly 1 hour.
Preheat the oven to 400 degrees F. Line a sheet pan with parchment paper.
On a floured board, roll the dough to a rectangle a little larger than 10-by-14 inches. Wrap the dough around your rolling pin and transfer it to the prepared pan. Using a ruler and a small knife, trim the dough to a 10-by-14-inch rectangle. Refrigerate for 15 minutes.
Remove the stem of each fig and cut them in quarters through the stem (or in sixths, if the figs are very large.) Place the figs in rows on the dough. Sprinkle with the full 1/2 cup of sugar and dot with the butter.
Bake for 50 minutes to 1 hour, until the pastry is browned and the fruit starts to brown. Rotate the pan once during baking. If the pastry puffs up in one area, cut a little slit with a knife to let the air out. Don't worry! The juices will burn in the pan but the tart (and the pan!) will be fine! When the tart's done, loosen it from the pan while it's still warm and transfer it to a board or clean piece of parchment paper.
In a small pan, heat the apricot jelly with 2 tablespoons water and brush the fruit and pastry completely with the mixture. Allow to cool, cut into squares, and serve warm or at room temperature.
Figs sold singly in trays will be larger than figs sold in pint baskets.