1 stick cold unsalted butter, cut into 1/2-inch pieces
1/4 cup solid vegetable shortening, chilled and cut into small pieces
4 to 5 tablespoons ice water
7 large Granny Smith apples—peeled, cored and cut into 1/4-inch dice
1 tablespoon fresh lemon juice
1 tablespoon dark rum
4 tablespoons unsalted butter
1/2 cup sugar
1/4 cup packed light brown sugar
1/2 cup all-purpose flour
4 tablespoons unsalted cold butter, cut into 1/2-inch dice
1/2 cup chopped pecans or almonds
In a food processor, combine the flour, salt and baking powder and pulse to mix. Add the butter and shortening and process just until the mixture resembles coarse meal. With the machine on, slowly add 4 tablespoons of the ice water and pulse just until the dough is evenly moistened and comes together; add more water if necessary. Transfer the dough to a work surface, then gather it into a ball. Pat the dough into a disk, wrap it in plastic and refrigerate for at least 1 hour.
In a large heatproof bowl, toss the apples with the lemon juice and rum. Melt the butter in a large skillet. Add the sugar and cook over moderate heat, stirring occasionally, until golden brown, 5 to 7 minutes. Add the apples and cook until tender, about 15 minutes. Return the apples to the bowl and let cool completely.
In a bowl, mix the brown sugar and flour. Using a pastry blender or 2 knives, cut in the butter until crumbly. Stir in the nuts.
Preheat the oven to 375°. On a lightly floured surface, roll out the dough 1/8 inch thick. Using a 6-inch pan lid as a guide, cut out 6 rounds. Transfer them to a baking sheet. Spread the apple filling evenly over each round to within 1 inch of the edge. Fold up or pinch the edges to hold the filling. Sprinkle the topping over the tarts and bake for 30 to 40 minutes, or until the crust is golden, the filling is bubbling and the tops are lightly browned. Let the tarts cool slightly on racks before serving.