Add the first 9 ingredients to a 1-1/2- or 2-pound bread machine according to the manufacturer's directions. Select the dough cycle. When cycle is complete, remove dough. Punch down. Cover and let rest for 10 minutes.
Divide the dough into thirds. On a lightly floured surface, roll each portion into an 18-inch-long rope. To shape, line up the ropes, 1 inch apart, on a greased large baking sheet. Starting in the middle, loosely braid by bringing the left rope under the center rope. Bring the right rope under the new center rope. Repeat to end. On the other end, braid by bringing the outside ropes alternately over the center rope to center. Press ends together; tuck under slightly.
Cover and let rise in a warm place for 35 to 45 minutes or until nearly double. Bake in a 375 degree F oven for 25 to 30 minutes or until bread sounds hollow when lightly tapped.
Remove from baking sheet; cool on a wire rack.
In a small bowl combine powdered sugar and vanilla. Stir in enough milk (1 to 2 tablespoons) to make an icing of drizzling consistency. Drizzle over bread.