Preheat oven to 375 degrees. Add water, milk, oil, sugar, dried cranberries, yeast salt, orange zest, orange juice, nutmeg, and 1 cup of the flour to a large mixing bowl. Stir to combine then let sit 5 minutes.
Add the next cup of flour, stirring to combine. Gradually add the third cup of flour until you can no longer stir with a spoon. Dust dough with additional flour and begin kneading. Knead, adding more flour as needed, until you have a soft, pliable dough that is no longer sticky. Let rise in a greased bowl, covered, for one hour.
Turn out dough onto a floured surface. Using a pastry cutter, divide dough into three parts and form into 10 inch ropes.
Pinch the three ropes together at one end and braid.
Whisk egg and milk together in a small bowl. Transfer braid to a sheet pan.
Brush braid with egg-milk mixture. Bake 25 minutes at 375 degrees. Cool on a wire rack.
Sift powdered sugar into a medium bowl. Stir in milk a little at a time until the icing reaches drizzling consistency. Drizzle over cooled braid.