Preheat the oven to 350 degrees F (175 degrees C). Grease an 8- or 9-inch square baking dish.
In a large bowl, cream together the shortening and sugar until smooth; mix in egg. Stir in the milk. Sift in the flour and baking powder, and mix well. Fold in the blueberries. Spread evenly in the prepared pan.
Bake for 50 to 60 minutes in the preheated oven, until a toothpick inserted into the center comes out clean.
In a medium bowl, beat the confectioners' sugar and butter together until smooth. Beat in the egg gradually, then stir in vanilla. Chill until serving. Serve chilled sauce over warm pudding.
