For crust: I recommend using Flakiest Pie Pastry or Dad's Pie Pastry, but any other pastry in this collection will do. Roll the dough into a 13-inch circle and line a 9-½-inch deep-dish pie plate with it, forming the overhanging dough into an upstanding rim. Place in the freezer for 15 minutes while you make the filling.
For filling: Combine the apples, 3 tablespoons of the sugar, and the lemon juice in a large bowl. Mix well. Set aside for 10 minutes to juice. Preheat the oven to 400ºF. After 10 minutes, mix the remaining sugar with the cornstarch and stir into the apples. Turn the filling into the chilled pie shell and bake on the center oven rack for 30 minutes. While the pie bakes, make the topping.
For topping: Process the pecans, sugar, and brown sugar in a food processor until the nuts are well chopped. Add the flour and salt. Pulse briefly, to mix. Scatter the cold butter over the mixture and pulse five or six times, until the mixture resembles coarse crumbs. Do not overprocess. Transfer the mixture to a bowl and gently rub it between your fingers until the texture is uniform. Refrigerate the topping.
After the pie has baked for 30 minutes, remove it from the oven and spread the crumb mixture evenly over the top. Tamp the crumb mixture gently, to even it out. Reduce the oven to 375ºF. Bake the pie for 30 to 35 minutes more, until the topping is nicely browned and any visible juices bubble thickly. Transfer the pie to a rack to cool. Serve warm or at room temperature, with some of the caramel sauce drizzled on top. Vanilla ice cream is in order here, too.
This confection will dress up any apple, pear, or peach dessert, especially if there is vanilla ice cream involved. Combine 1 cup of packed light-brown sugar, 1 cup of heavy cream, and ¼ cup (½ stick) of unsalted butter in a heavy, medium saucepan. Boil for 8 minutes, stirring often. Pour into a heatproof bowl. The sauce will thicken as it cools. Serve warm. Makes about 2-½ cups, or 8 servings.