In a heavy, medium saucepan, combine the half-and-half, chocolate, granulated sugar and cornstarch. Bring just to a boil over medium heat and cook, whisking, until thickened, about 1 minute. Remove from the heat and stir in the butter. Let cool slightly, then press plastic wrap onto the surface to cover. Refrigerate until cold, at least 2 hours or overnight.
In a large bowl, toss together the flour, baking powder and salt. Using your fingers or a pastry cutter, mix in the shortening until coarse crumbs form. In a small bowl, beat together the whole egg and ice water. Stir into the flour mixture with a fork until the dough comes together. Divide into 6 even balls.
On a lightly floured surface, roll each ball of dough into a 6-inch-wide disk. Brush a 1/2-inch border of the beaten egg white around each rim. Spoon 1/3 cup of the chilled pudding in the center of each disk, spreading slightly. Fold the dough over the filling to cover, press the edges to seal, then crimp with a fork. Transfer to a parchment-paper-lined baking sheet and refrigerate until ready to fry.
In a deep, heavy 9- to 10-inch skillet, heat the oil over medium-high heat until it registers 375 degrees on a deep-fry thermometer. Add 2 pies at a time and fry, turning 2 to 3 times, until deep golden, about 3 minutes. Let cool for 15 minutes, then dust with the confectioners' sugar.