Add warm milk, warm water and egg to bread pan. Add flour, sugar, and salt to pan. Tap pan to settle dry ingredients, then level ingredients, pushing some of the mixture into corners. Place shortening into corners of pan.
Make a well in the center of the dry ingredients; add yeast. Lock pan into bread maker.
Use the DOUGH bread select setting on your bread machine. Start bread maker. When done, remove pan from bread maker.
Place dough onto floured surface and let rest for 15 minutes.
Roll dough into a rectangle about 1/2-inch thick. Cut with doughnut cutter. Leave on counter to rise, covered.
Knead together scraps of dough, let rest 10 minutes and reroll. Cut into doughnuts. Let doughnuts rise, covered, in warm, draft-free place for 30-45 minutes or until doubled in size.
Fry doughnuts in 3 inches of hot oil. They'll cook quickly! Turn doughnuts over as they rise to the surface. Fry until golden brown on both sides. Remove from oil and place on paper towel to drain.
Combine sugar and vanilla in mixing bowl. Stir in enough milk for icing consistency.
Dip doughnuts one a a time in icing and place on raised wire rack to dry.
Doughnuts are best when still warm.