Preheat oven to 170C/325F. Line or grease a muffin tray.
Peel and finely grate the carrots. Set aside.
In a large bowl, cream together the butter and sugar until fluffy. Beat in the eggs, one at a time, and then beat in the carrot, vanilla and orange zest.
Sift in the flour, baking powder and salt and stir to gently combine.
Fill the muffin cups ¾ full and bake for 15-20 minutes or until an inserted skewer comes out clean. Allow to cool in the tins for 10 minutes then remove and allow to cool completely on a wire rack. Frost with the icing when cold.
For the icing:
In a large bowl, cream the butter until fluffy. Sift in the icing sugar and beat until thick. Add enough of the orange juice to make it a frosting consistency.