Preheat the oven to 350 degrees F. Butter a 9-by-13-inch baking dish, line the bottom with parchment paper and butter the parchment. Dust with flour and tap out any excess. Whisk the flour, cocoa powder, gingerbread spice and salt in a medium bowl. Beat 1 stick butter and the brown sugar in a large bowl with a mixer on low speed until incorporated, then increase the speed to medium and beat until fluffy, about 5 minutes. Add the whole egg and egg yolks and beat 1 minute. Reduce the mixer speed to low and add the flour mixture in two batches, alternating with the water. Beat until just smooth, being careful not to overbeat (the batter may look curdled).
Spread the batter in the prepared pan. Bake until the cake springs back when pressed, 25 to 35 minutes. Let cool 5 minutes in the pan, then turn out onto a rack, remove the parchment and let cool completely.
Beat the remaining 4 tablespoons butter and the ginger jam in a large bowl with a mixer on medium-high speed until pale, about 1 minute. Gradually beat in the confectioners' sugar until light and fluffy, about 2 minutes.
Carefully cut the cake in half crosswise with a serrated knife and spread one half with the ginger filling. Top with the other cake layer and cut lengthwise, then crosswise into 6 equal rows, making small cubes. Freeze the cubes until firm, about 1 hour.
Melt the chocolate and shortening in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water), about 5 minutes. Remove the bowl from the pan. Hold each cake square over the bowl on a fork and spoon chocolate over the top until evenly covered. Let the excess drip off, then transfer to a rack set on a baking sheet. Top with crystallized ginger. Let set, about 2 hours at room temperature or 30 minutes in the refrigerator; before completely set, loosen with a spatula.