Add first 8 ingredients to a 2-pound bread machine according to the manufacturers directions. Select the dough cycle. When cycle is complete, remove dough. Punch down. Cover and let rest for 10 minutes.
Meanwhile, for filling, place apples in blender container. Cover; blend until nearly smooth. Mix egg white and water; set aside.
Divide dough in half. On a lightly floured surface, roll each half of the dough into a 12-inch square. Cut into three 12x4-inch strips. Spread about 2 tablespoons filling down center of each strip. Brush some egg white mixture on edges. Bring long edges together over filling and pinch to seal. To shape, line up 3 of the filled ropes, seams down, 1 inch apart on a greased baking sheet. Starting in middle, loosely braid by bringing left rope under center rope. Bring right rope under new center rope. Repeat to end. On other end, braid by bringing outside ropes alternately over center rope to center. Press ends together; tuck under. Repeat with remaining ropes to make a second braid. Brush with more egg white mixture. Cover; let rise for 45 to 60 minutes or until nearly double. Brush with remaining egg white mixture. Bake in a 350 degree F. oven about 30 minutes or until bread sounds hollow when tapped. If necessary, cover loosely with foil last 10 to 15 minutes to prevent overbrowning. Remove; cool on wire rack.
Mix powdered sugar and vanilla. Stir in enough milk (3 to 4 teaspoons) to make a glaze of drizzling consistency. Drizzle over braids. Makes 2 loaves (22 servings.)