First, make the Sugar Dough: In a food processor fitted with the metal blade, combine the flour and sugar. Add the butter and process until the mixture resembles fine meal. In a small bowl, whisk the egg yolks with 1 tablespoon of the cream. Scrape into the processor and process until a ball begins to form, adding more cream if necessary to bring the dough together. Remove the dough and, on a lightly floured surface, press it down into a circle. Wrap in plastic wrap and refrigerate for at least 1 hour.
To form the crust, divide the pastry into two parts. Wrap one in plastic wrap and set aside. On a lightly floured surface, roll the other piece into a round large enough to line and overlap a 10-by-2-inch pie plate. Transfer to the pie plate and, leaving a 1/2-inch overhang, trim the edge with a sharp knife. Tuck the overhang under itself to make a thicker rim. Refrigerate for at least 30 minutes. (If using a glass pie plate, remove 15 minutes before filling.)
To prepare the filling, peel, core, and quarter the apples, then cut into 1/4-inch slices. Divide the butter among 2 or 3 large skillets and melt it over medium heat until it just begins to brown and smell nutty. Divide the apples among the skillets and stir gently to coat with butter. Saute over medium heat until lightly browned and tender, 15 to 20 minutes, stirring often. Transfer the apples to a large baking sheet and spread out evenly to cool.
Preheat the oven to 400 degrees. In a small saucepan, warm the Calvados over medium-low heat. Combine the prunes, apricots and figs in a small bowl and pour the Calvados over. Leave to plump for 15 minutes, then drain off any excess liquid.
In a large bowl, stir together the apples, dried fruit, honey, lemon juice, cinnamon, zests and nutmeg. Spoon the filling into the prepared pie plate. On a lightly floured surface, roll out the remaining dough to a 9-inch circle and transfer to the pie. Pinch the edges together. With the tip of a sharp knife, cut several evenly spaced 1-inch vents in the top pastry. Brush with egg white and sprinkle with sugar.
Bake the pie for 30 minutes. Turn down the oven to 350 degrees F and bake until the crust is golden brown, 35 to 40 minutes more. Transfer to a wire rack to cool before cutting into wedges and serving.