Preheat oven to 325°F and line light baking pans with parchment paper.
In a large bowl with a handheld mixer, or stand mixer fitted with the paddle attachment, beat the butter and powdered sugar on high speed until smooth and creamy, 1-2 minutes. Add the orange zest, orange juice, and vanilla and beat until combined, scraping down the sides of the bowl as needed.
1 cup unsalted butter, 3/4 cup powdered sugar, Zest from 2 oranges, 1 Tablespoon fresh orange juice, 1/2 teaspoon vanilla extract
With the mixer on low, add the flour, cornstarch, and salt and beat until it comes together, then turn it up to medium and beat until it starts to come together and pull away from the sides of the bowl.
2 cups all-purpose flour, 1 Tablespoon cornstarch, 1/4 teaspoon salt
Dust a piece of parchment paper or clean counter with flour then turn the dough out onto it. Using a floured rolling pin, roll the dough out to 1/2-inch thick circle/rectangle.
Use a 2-inch cookie cutter to cut out the dough. Re-roll and cut the dough until it’s all used.
Arrange the cookie dough 1-2 inches apart and bake for 10-12 minutes, or until the edges are set and very light golden. Allow cookies to cool on the baking sheet for 5 minutes then transfer to a wire rack to cool completely before icing.
To make the icing, whisk together the powdered sugar, milk, and orange juice. Dip the tops of the cooled cookies into the icing and place them back on the cooling rack. Sprinkle with orange zest and coarse sugar if desired then let it set for 30 minutes to 1 hour.
1 cup powdered sugar, 1 Tablespoon fresh orange juice, 1 Tablespoon milk