In electric mixer, beat together butter and sugar until combined. Beat in egg. Add flour until combined.
Roll half of the dough at a time out onto a floured surface until about 1/8" thick. Cut with star-shapped cookie cutter. Reroll trimmings and cut again. Place on ungreased baking sheet. Bake 6-7 minutes. Transfer to wire grid to let cool completely.
Glaze cooled cookies with white or tinted powdered sugar glaze. Before it sets, dot icing with contrasting color glaze and pull a toothpick through to create marbling. Let dry completely.
Store in airtight container, using waxed paper to separate layers.