Prepare Dough

1. Place the flours, sugar, and salt in the bowl of a food processor fitted with the steel blade. Cover and pulse 2-3 times to mix.
2. Add the cut-up shortening and butter and process 5-10 seconds until the dough is the size of small peas.
3. Add 4 tablespoons of cold water and pulse, 3-4 times. Watch the dough carefully and stop the machine as soon as the dough starts to clump together. It will look rough and lumpy, and there may be pieces of butter. If it looks too dry and crumbly, sprinkle in more water and pulse 1 more time. Pinch the dough together with your fingers; if the dough holds together, it is done. Do not allow the dough to form a ball on the machine blades as it will be overworked and tough.
4. Turn the dough out onto a piece of plastic wrap. Form the dough into a flat disc around 5 to 6 inches in diameter, wrap with plastic wrap, and refrigerate for at least 30 minutes (up to 2 days).
5. Take the dough out of the refrigerator and let it sit at room temperature for around 5 minutes (this makes it easier to roll).
6. On a lightly floured work surface, roll the dough into a circle 1/2 inch larger than the diameter at the top edge of the skillet you will use to cook the tarte. Place rolled dough on a waxed paper-lined cookie sheet and put back in the refrigerator.

Prepare Filling

1. Move an oven rack to the upper-middle position and preheat the oven to 375 degrees.
2. Peel the apples, cut them in half pole to pole, then remove the core and stem (I use a melon baller). Cut each piece in half; you should have a total of 24 to 28 apple pieces. As you prepare the apples, place them in a large bowl and toss with the lemon juice.
3. Melt 8 tablespoons butter in a 9 or 10-inch ovenproof skillet, preferably cast iron, over medium-low heat. Sprinkle the sugar evenly over the top of the butter; swirl the pan to combine.
4. Turn off the heat and arrange the apple pieces over the sugar in a circular pattern. First, make a circle of apple slices just inside the pan wall, positioning the apple slices on their side. Fit apple pieces tightly together, overlapping them. Next, fill the middle with apple slices to cover the entire surface.
5. Place the skillet over medium-high heat and cook until the juices turn a butterscotch/amber color, around 12-15 minutes.
6. Remove from heat and, using 2 forks, gently turn the apple quarters over so the caramelized side is on top and the outside (round side) of the apple pieces is facing down. Be careful as the skillet is very hot, and the apples and sauce will be steamy!
7. Remove the rolled dough from the refrigerator.
8. Return the skillet to medium heat and continue cooking until the sauce is a darker amber color, about 4-5 minutes more. Remove from heat.

Bake

1. Fold the rolled dough in half, then in quarters. Center the point over the center of the apples; unfold the dough over the apples. Using a fork, carefully and quickly tuck the dough in around the outside of the apples. Tuck in any extra dough to add strength to the sides. Cut 4-5 air holes in the dough, around 1-2 inches from the center.
2. Bake until crust is golden brown, around 25-30 minutes. Remove skillet from oven and set on a wire rack to cool.
3. After around 5-10 minutes, if the bubbling around the edges of the tarte has stopped, and the caramel looks thick, syrupy, not watery (peek inside one edge), it’s time to flip it out of the pan.
4. Run a knife around the edge of the tarte to loosen. Place a large rimmed serving plate over the skillet and, using potholders, carefully flip the tarte over. Remove the skillet. Using a spoon, remove any apples that stick to the skillet and place them back in place on the tarte. Spoon any remaining caramel from the pan over the top of the tarte.
5. Let the tarte rest for 45 minutes before cutting and serving. It can rest at room temperature for up to 5 hours; after that, if needed, rewarm in a 300-degree F oven for 5-8 minutes to gently relax the caramel sauce.
6. Serve with vanilla or butter pecan ice cream.