Butter and flour a 9-inch glass pie plate. On a lightly floured surface roll the dough to a 12-inch round about 1/8-inch thick. Line the pie plate, being sure to press the dough into the bottom and up the sides without stretching. Chill 30 minutes. Roughly chop the apples into small pieces, about the size of lima beans. (The apples should measure about 9 cups chopped.) Combine with the sugar, tapioca, cinnamon, salt and lemon juice in a large bowl. Set aside until the juices begin to run, about 10 - 15 minutes. Beat together the cream and egg to make a glaze and brush over the edges and bottom of the crust. Mound the apple filling in the pie shell. It will be quite full. Roll out the remaining dough to a large round about 1/8'' thick. Place over the filling a seal the edges together by gently pressing together. Trim any excess dough with scissors and flute the edges. Brush the top with the remaining egg glaze. Using a paring knife, cut out and remove a circle the size of a quarter from the center of the top crust. Cut 5 or 6 slits in a spoke pattern beginning 1/2-inch from the center hole and ending 1/2-inch from the outside edge. Chill for 30 minutes before baking.
Preheat oven to 425 degrees. Place the pie plate on a cookie sheet and bake 10 - 15 minutes, or until the top is golden brown. Reduce the heat to 300 degrees and continue baking until the juice visible in the center becomes thick and bubbly, about 1 hour (if the edges are browning too quickly, cover with aluminum foil). Meanwhile prepare the caramel topping. Spoon the hot caramel over the top of the pie to coat and return to the oven. Bake 5 minutes longer, or until the caramel sets. Cool on a rack before serving.
In a small saucepan combine brown sugar, butter and cream. Bring to a boil and cook for 2 minutes. Remove from the heat and stir in the pecan halves.
In a large bowl combine the flour with the lard, butter and salt. Mix lightly with your fingertips until the dough forms grape-sized pieces. You should be able to see chunks of fat. Stir in the iced water. Lightly knead, handling the dough as little as possible, until the dough forms a ball. Add a little more iced water if necessary. Transfer to a plastic bag and shape into a log. Seal the bag, pressing out any air, and chill for as little as 1 hour or as long as 3 days. The pie dough can be frozen for as long as a week.