Roll out dough. Cut out pattern; bake and cool hat cookie. Outline hat using tip 3 and full-strength royal icing. Thin icing and fill in hat with tip 3; let dry. When completely dry, use tip 3 to outline and pipe-in hat band and buckle; attach shamrock icing decorations. Set aside.
Bake and cool cake. Trim off ears; reserve. Outline facial features with tip 3. Cover face with tip 16 stars. Build up nose with additional tip 16 stars. Pipe tip 16 swirls for hair and beard. Position hat on head supporting with reserved cake pieces.