Preheat oven to 350°F. Line two baking sheets with parchment.
Cream butter and sugar in a bowl until pale and fluffy, about 2 minutes.
Beat in vanilla, orange zest, and milk until smooth.
Whisk flour, cornmeal, baking powder, and salt. Add to the bowl and mix just until a soft dough forms.
Dust the counter lightly with flour. Roll dough to about 1/4 inch thick.
Cut stars with a 2-inch cutter. Gather scraps and re-roll once to use up dough.
Bake 10–12 minutes until edges look set and lightly golden. Rotate pans halfway if needed.
Cool on the sheet 5 minutes, then transfer cookies to a rack to cool completely.
Whisk confectioners' sugar, orange juice, and orange zest until glossy and drizzleable.
Drizzle or dip cooled cookies in the glaze. Let set until the glaze firms.