Make the cookie dough: In the bowl of a stand mixer fitted with the paddle attachment, cream the butter, sugars, salt, orange zest, and vanilla on medium speed until light and fluffy, 4 to 5 minutes.
Add the egg, scraping down the bowl after it's fully incorporated.
Reduce the speed to low. Add the flour all at once and mix just until combined, about 1 minute.
On a lightly floured surface, roll out the dough to a square about 12 12 inches.
Make the cinnamon smear: In the bowl of a stand mixer fitted with the paddle attachment, mix all the ingredients on medium speed until smooth, about 2 minutes. Scrape down the bowl as needed.
Spread the smear on the dough and roll up into a tight log. Wrap the log in plastic wrap and freeze until firm, 15 to 20 minutes.
While the dough is firming up, preheat the oven to 375 degrees F. Line two baking sheets with parchment paper.
Using a sharp knife, cut the log into 1/4-inch-thick slices. Transfer the cookies to the prepared baking sheets, spacing them about 1 1/2 inches apart.
Bake until golden around the edges, 12 to 14 minutes. Rotate and switch the baking sheets as necessary for even baking.
Allow the cookies to cool for a minute on the baking sheets then transfer, using a spatula, to a cooling rack and allow to cool completely.
Make the icing: Combine all the ingredients in a small bowl. Lightly brush a thin layer on the surface of each cooled cookie. Let set before serving.
Store in an airtight container.
Reprinted with permission from Cookies at Home with the Culinary Institute of America, by The Culinary Institute of America, copyright (c) 2011. Published by Wiley.