Place the warm water in a mixing bowl. Sprinkle the yeast evenly over the water to dissolve. When it has bubbled (proofed), add the milk, sugar, salt, egg, shortening, and 1 cup of the flour. Beat the mixture on medium speed, scraping down the sides of the bowl as needed, for about 2 minutes. Stir in the remaining flour by hand until the dough is smooth. Cover the dough with a damp towel and let rise in a warm place until doubled in size, about 1 hour. Turn the dough out onto a floured work surface. Pat it flat then roll it out with a rolling pin until it is 1/2-inch thick. Cut the dough with round cookie cutters. Place the cutouts, evenly spaced, on floured baking sheets and let rise in a warm, draft-free place until doubled in size, about 30-40 minutes. While the doughnuts are rising, make the vanilla filling. In a mixing bowl, cream together the shortening and butter until smooth and blended. Gradually add the confectioners' sugar (1/2 cup at a time) with the mixer running. Add the vanilla and milk and beat the filling until light and fluffy. The filling can be stored up to 2 weeks in an airtight container in the refrigerator. When the doughnuts have risen, heat the vegetable oil to 350 degrees F in a deep fryer or Dutch oven. Carefully slide the dough into the hot oil. Turn the doughnuts as they brown, they should cook for about 1 minute per side. Do this in batches as needed making sure to keep the oil at 350 degrees F. Carefully remove the browned doughnuts from the oil with a spider or slotted metal spoon. Set on a rack to drain. When the doughnuts are cool, using a sharp, narrow knife, carefully create a pocket inside the doughnut to hold the filling. The opening to the pocket should be as small as possible (so the filling won't leak out). Fill a pastry bag with a plain tip with the vanilla filling. Insert the tip into the hole of the doughnut and gently, but generously, fill with the cream filling. Dust the filled doughnuts with additional confectioners' sugar.