Preheat the oven to 325°F (163°C). Line two baking sheets with parchment paper.
In a large bowl, beat the softened unsalted butter and powdered sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the key lime zest, key lime juice, and vanilla extract to the butter mixture and mix until combined.
In a separate bowl, whisk together the all-purpose flour and salt.
Gradually add the flour mixture to the wet ingredients, mixing on low speed until just combined. The dough will be soft but manageable.
Turn the dough out onto a lightly floured surface and gently knead it a few times to bring it together.
Roll the dough out to about 1/4 inch thickness.
Using a 2-inch round cookie cutter, cut out cookies and place them on the prepared baking sheets about 1 inch apart.
Sprinkle the tops of the cookies lightly with granulated sugar.
Bake in the preheated oven for 18 to 20 minutes, or until the edges are just beginning to turn golden.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.