Step 1Heat the oven to 325 degrees. Line 2 baking sheets with parchment paper.
Step 2In a large bowl, combine one-fourth cup powdered sugar, the butter, sesame seeds, 2 tablespoons za’atar, the salt and vanilla and beat with an electric mixer on medium speed until whipped and light, about 3 minutes (see Baker’s Note). Add the flour and mix on low speed until just combined.
Step 3Using a half-ounce ice cream scoop or 1 tablespoon, portion the dough and roll each portion into a smooth ball before arranging 1 inch apart on the prepared baking sheets.
Step 4Bake, rotating the baking sheets from front to back and top to bottom halfway through, until the cookies are lightly browned on the bottom and set, about 25 minutes.
Step 5While the cookies bake, whisk together the remaining 2 cups powdered sugar and 2 tablespoons za’atar in a large bowl. Immediately after the cookies come out of the oven, work in batches to gently toss them in the za’atar sugar until completely coated. Return them to the baking sheets and let cool completely.
Step 6Once all the cookies are cooled, toss them again in the za’atar sugar until completely coated. When serving, arrange the cookies on a plate and sprinkle with more za’atar, if you like.