Make the pastry: in a large bowl or the bowl of a food processor, stir together the flour, sugar and salt. Add the butter, shortening and cheese and use a fork, pastry blender, wire whisk or the “pulse” motion of the food processor to blend the mixture until it resembles coarse meal, with lumps of fat no bigger than a pea.
Drizzle the minimum amount of water over the mixture and stir until the dough comes together, adding a little more a bit at a time if you need it. Gather the dough into a ball, flatten it into a disc, wrap it in plastic and chill for at least half an hour. On a lightly floured surface, roll the dough out into a 14-inch circle (don't worry about it being perfect); drape over the rolling pin and transfer to a parchment-lined rimmed baking sheet. Put the fridge while you make the filling.
Meanwhile in a large, heavy skillet, heat the butter over medium-high heat. When the foaming subsides, add the apples and cook until they start releasing their juices. Sprinkle the sugar and salt overtop. Cook for about 10 more minutes, stirring often, until the moisture evaporates and the syrup thickens, and the apples start to turn golden. Remove from the heat and transfer to a bowl to cool.
To make the crumble topping, blend all the ingredients in a bowl or food processor until well combined and crumbly. Preheat the oven to 375F.
Pile the cooled apples onto the pastry, mounding it the middle and spreading it out to within 2 inches of the edge. Fold the pastry over, letting it fold where it wants to. If you like, brush the folded-over part with a little milk or beaten egg and sprinkle with coarse sugar. Scatter the crumble mixture over top.
Bake for 45 minutes, or until bubbly and golden. Let cool for at least 10 minutes before sliding carefully onto a cutting board to serve.