On a lightly floured work surface, roll out half of the dough into a 16-inch square. Cover the remaining half with plastic wrap; set aside. In a small bowl, combine ginger, cinnamon, coriander, nutmeg, and mace. In a large bowl, combine dried fruit.
Spread 2 ounces shortening over top of dough, leaving a 1/2-inch border around perimeter. Sprinkle 6 tablespoons turbinado sugar over shortening. Sprinkle one-fourth of combined spices over sugar. Sprinkle one-fourth of dried fruit over spices. Using the palms of your hands, gently press fruit into dough.
Fold four corners of square into center, creating a smaller square, enclosing filling. Gently press down on dough with rolling pin. Spread 2 ounces shortening over dough, leaving a 1/2-inch border around perimeter. Sprinkle 6 tablespoons sugar, another one-fourth of spices, and another one-fourth of fruit. Using palms of your hands, gently press fruit into dough; fold dough into thirds.
Transfer dough to a piece of parchment paper; roll dough out into an 11-by-17-inch rectangle. Carefully lifting parchment, transfer to baking sheet. Let stand about 20 minutes.
Repeat steps one through four with remaining dough, shortening, turbinado sugar, spices, and dried fruit.
Preheat oven to 350 degrees. Combine jam and cognac in a small saucepan. Set over low heat; cook until liquefied. Strain jam, discarding solids. Brush surface of lardy cakes with jam.
Bake cakes until golden and puffed, 35 to 45 minutes. Let cool slightly before serving. Lardy cake will keep for up to 1 day, wrapped in aluminum foil; reheat cake before serving.