Beat the butter in the bowl of a mixer until light yellow and creamy. Add 2 1/2 cups confectioner’s sugar and vanilla and continue beating until smooth and light.
Sift 1 cup confectioner’s sugar into a small bowl. Whisk in the orange liqueur and milk until smooth. Put the icing into a sandwich baggie. Give it a twist to close it and cut a small hole in one corner. Drizzle the icing across the shortbread pieces. Sprinkle with demerara sugar.