In a medium bowl, use an electric mixer to cream the butter and sugar together.
In a small bowl, whisk together the flour, salt and baking powder. Set aside.
Stir lemon zest into the butter and sugar mixture. Then stir in the dry ingredients.
Pat the mixture into an 8-inch round or square pan. Bake for 20 to 25 minutes.
ICING
While the shortbread is baking, prepare icing. Mix all ingredients together, adding lemon juice until a desired consistency is reached. If it ends up too soupy, thicken it up a bit with more powdered sugar. Too thick? Add more lemon juice. It should be just thin enough to drizzle onto the shortbread, but not so thin that it won't set.
Drizzle on shortbread while it's still warm. Let cool completely and refrigerate until icing is set.