1. Prepare the orange sugar: In a small bowl, combine the coarse sugar and the orange zest. Use your fingers to massage the zest into the sugar until very fragrant. Spread the sugar in a single layer on a plate and let stand at room temperature, uncovered, to dry while you make the cookies.
2. Prepare shortbread cookies: Line 2 baking sheets with parchment paper or nonstick baking mats.
3. In the bowl of a stand mixer fitted with the paddle attachment, combine the butter, icing sugar and salt. (You can also use an electric hand mixer and large bowl.) Beat on medium speed until smooth and fluffy, 2 minutes. Turn off the mixer and add the flour. Mix on low speed until a dry dough forms, 2 to 3 minutes.
4. Transfer the dough to a piece of parchment paper lightly dusted with flour and press into a ball. Sprinkle the top of the dough with flour, place a second sheet of parchment paper on top and roll the dough out until it is ½ inch thick. Remove the top piece of parchment paper and use a 2½-inch round cookie cutter to cut out rounds and place on the prepared baking sheets, spacing them about 1½ inches apart. Press the scraps of dough together to form a ball and repeat the rolling and cutting process. Freeze the cut cookies for 20 minutes.
5. Preheat oven to 350°F and place 2 racks in the upper middle and lower middle positions. Bake both sheets at the same time, switching them halfway through, until the tops of the cookies are lightly browned, 15 to 19 minutes. Let cool completely on baking sheets.
6. Prepare the glaze: In a large bowl, whisk together the Campari, icing sugar, vanilla extract and salt until smooth. (If frosting is too thick, add more Campari; if too thin, add additional icing sugar.) Dip tops of cookies in frosting, letting excess drip into bowl. Sprinkle the cookies with the orange chips and let stand for 1 hour.
Storage: Cookies will keep in an airtight container in a single layer for up to 1 week. (Just make sure the glaze is completely dry first.)