Add the butter, a ½ cup of the powdered sugar, and the salt to a stand mixer fitted with the paddle attachment. Mix on medium, then medium-high for 2 minutes, scraping the inside of the bowl halfway through, until the butter and sugar are combined and smooth.

Mix in 1 teaspoon of the lime zest, the lime juice, and vanilla. Add the flour and mix for about 90 seconds on medium-high until a ball of dough forms in the bowl.

Roll the dough into balls about the size of a key lime. Flatten slightly and arrange on the baking sheet, 10 to 12 cookies on the pan. Bake 10 to 12 minutes until firm and the bottom edges just begin to brown.

While the cookies bake, stir together the remaining ¾ cup powdered sugar and 1 teaspoon of lime zest in a shallow bowl.

Remove the cookies from the oven and let cool for 5 minutes. Roll each cookie to coat it in the powdered sugar and set them on a cooling rack over a sheet pan to cool completely.

Repeat the process with the remaining cookie dough.

Once all the cookies are rolled in powdered sugar, place the remaining powdered sugar and zest in fine mesh sieve or strainer and dust the tops of the cookies with any remaining sugar.

Let cool completely and store in an airtight container for up to 5 days.