In a bowl, cream together butter and sugar. Add vanilla and lime zest and stir to combine.
In a separate bowl, sift (or whisk) together flour and salt. Add dry ingredients to butter mixture and mix until a dough forms. Dough may look crumbly, but will hold together when pinched between two fingers.
Shape into a long, round tube about 2 inches thick and wrap in wax paper. Refrigerate for 30 minutes.
Meanwhile, preheat oven to 350 degrees. Line a baking sheet with parchment paper and set aside.
Remove dough from refrigerator and unwrap. Using a sharp knife, slice dough into ¼ inch slices and place on lined baking sheet.
Bake for 10-12 minutes or until cookies barely begin to brown around edges. Remove from oven and allow to cool for 2-3 minutes before removing from sheet to cool completely.
While cookies are cooling, combine confectioner's sugar, lime juice, and lime zest. If too thick, add more lime juice 1 teaspoon at a time.
Once cookies have cooled completely, ice cookies with a knife and set on a cooling rack for one hour until set.
Your cookies will keep for 5-7 days in a sealed container. To help keep cookies fresh, you can place a slice of bread in the container with the cookies. Replace the bread slice as needed.
You can keep raw cookie dough in the refrigerator for 3 days. Cookie dough and baked cookies freeze well, up to 3 months in a sealed plastic container or freezer bag. Allow frozen dough to thaw for up to 10 minutes on the baking sheet before baking, or add 1-2 minutes to the baking time.