Add first 8 ingredients to a 1-1/2- to 2-pound bread machine according to the manufacturer's directions. Select dough cycle. When the cycle is complete, remove dough from machine. Punch down. Turn dough out onto a floured surface. Divide in half. Cover and let rest for 10 minutes.
Roll each half of dough to 1/2-inch thickness. Cut with a floured doughnut cutter. Reroll and cut trimmings. Cover and let rise for 45 to 60 minutes or until light. Heat cooking oil to 365 degrees F. Carefully fry doughnuts 2 or 3 at a time, about 2 minutes, turning once. Drain on paper towels.
In a medium bowl combine powdered sugar, vanilla, and enough milk ( 2 or 3 tablespoons) to make of drizzling consistency. Drizzle over doughnuts.