In a large bowl (or food processor), mix the sugar and lemon zest and gently rub together to help release all those lovely citrussy oils. Add the plain flour, desiccated coconut, baking powder and cold cubed butter; using a pastry cutter or your fingertips, rub the butter in until the mixture looks like fine breadcrumbs.
Bring the dough together with your hands—the warmth will help it bind. Divide the dough into 3 equal pieces. Roll each piece into a log (about 25-30cm long). Place the logs on 2 baking trays lined with parchment paper and flatten them slightly with your palm or a rolling pin so they are roughly 5-6cms wide and 1/2cm thick.
Bake in the centre of a preheated oven at 180°C for about 8-12 minutes. You want them to be just set and very lightly golden at the edges, but still pale.
While the logs are baking, whisk the icing sugar with the lemon juice and a tiny splash of boiling water to make a thick, spreadable icing.
As soon as they come out of the oven, brush the icing over the hot logs then sprinkle over the zest. These are cut in a similar way to my recipe for Brune Pinner: wait just a minute for the icing to set slightly, then use a sharp knife to cut the logs diagonally into 2cm wide slices while they are still warm. If you wait until they are cold, the shortbread will crumble and break!
Let the slices cool completely on the tray. The icing will become crisp and "crackly," and the biscuits will firm up beautifully.