In the bowl of a food processor fitted with a steel blade, place the flours, sugar and butter. Pulse 5 or 6 times in 1/2-second bursts to break up the butter.
Add the shortening, turn on the machine and immediately add the ice water, pulsing 2 or 3 times. The dough should look like a mass of smallish lumps and should just hold together in a mass when a handful is pressed together. If the mixture is too dry, pulse in more water by droplets.
Turn the dough out onto the work surface and with the heel of your hand, rapidly and roughly push egg-size blobs into a 6-inch smear. Gather the dough into a relatively smooth cake, wrap in plastic and refrigerate at least 2 hours (or up to 2 days).
Slice the halved apples into 4 lengthwise wedges each, and toss in a large bowl with the lemon juice and zest and 1/2 cup sugar.
Drain the apples after macerating 20 minutes.
In a 9-inch skillet melt the butter over high heat. Stir in the remaining sugar and cook until the syrup bubbles and caramelizes, and turns a brown color.
Remove the pan from the heat and arrange a layer of apple slices in a neat pattern on the caramel in the skillet, then arrange the remaining apples neatly on top.
Return the pan to moderately high heat and cook for about 25 minutes, covering the pan after 10 minutes. Every few minutes press down on the apples and baste them with the exuded juices. When the juices are thick and syrupy, remove the pan from the heat.
On a lightly floured surface, roll the chilled dough into a circle, 3/16-inch thick and 1-inch larger than the top of the pan. Drape the dough over the apples, pressing the edge of the dough between the apples and the inside of the pan.
Cut 4 small steam holes on the top of the dough.
Bake until the pastry has browned and crisped, about 20 minutes.
Unmold the tart onto a serving dish (so the pastry is on the bottom), and serve warm or cold with whipped cream or vanilla ice cream, as desired.
