Preheat oven to 325 degrees F. Lightly grease a doughnut pan.
In a small bowl, combine sugar and lime zest, pressing with the back of a spoon to release the oils from the zest so the flavor incorporates throughout the sugar.
In a large bowl, mix flour, sugar & lime mixture, baking powder, and salt.
In a medium bowl, combine in milk, lime juice, eggs, vanilla and melted butter. Beat together until well blended.
Add wet ingredients to dry ingredients, and stir until completely combined.
Use a pastry bag fitted with a large round tip (or a Ziploc bag with a corner cut off) to pipe the batter into a 6-doughnut pan. Bake 8 to 10 minutes in the preheated oven, until doughnuts spring back when touched. Allow to cool slightly before removing from pan.
To prepare icing, combine all icing ingredients in a small bowl. Stir until well incorporated.
Once doughnuts have cooled, dip them into icing or spoon icing over the doughnuts. Top with lime zest and shredded coconut.