Dissolve yeast in warm water in mixing bowl. Add milk, sugar, salt, egg, shortening and 1 cup of flour.
Beat on low speed scraping bowl constantly, 30 seconds. Beat on medium speed, scraping bowl occasionally, 2 minutes.
Stir in remaining flour until smooth. Cover and let rise in warm place until double, approximately 50-60 minutes.
Turn dough onto floured surface. Roll dough 1/2-inch thick.
Cut with round cookie cutter. Cover and let rise on floured baking sheets until double, 30-40 minutes.
Heat vegetable oil in deep fryer or Dutch oven to 350 degrees F. Slide doughnuts into hot oil. Turn doughnuts as they turn golden brown, about one minute on each side. Remove carefully from oil (do not prick surfaces) and drain.
When cool, make small hole to insert vanilla filling (recipe follows) . Take a sharp narrow knife and carefully make a large cavity inside of the doughnut to hold the filling. Fill the doughnuts generously and dust heavily with confectioners' sugar.
Vanilla Filling: Cream butter and shortening, gradually add sugar 1/2 cup at a time. Add milk and vanilla, beat until light and fluffy. Will keep for 2 weeks in airtight, refrigerated container.