1. Preheat the oven to 350 F and line two large baking sheets with parchment paper. In a large mixing bowl, sift together the flour, baking soda and salt. Gently whisk in the key lime zest.
2. In the bowl of a stand mixer fitted with the paddle attachment, mix the butter on high for about a minute until light and fluffy. Add the sugar and mix again for another minute. With the mixer running on low, add in the egg, followed by the vanilla extract. Scrape down the sides with a rubber spatula as needed. Add in the key lime juice. With the mixer on low, add in the dry ingredients to the wet until just combined. Stir in the graham cracker crumbs by hand.
3. Using a medium cookie scoop (about 1.5 tablespoons) scoop the cookie dough on to the prepared baking sheets, a few inches apart. Bake the cookies for 9-10 minutes, or until the edges are just set and the inside is still soft. Remove the cookies from the oven and allow them to cool for 10 minutes on the pan, before removing them to a wire rack to cool completely.
4. Make the icing. Whisk together the powdered sugar, heavy cream, key lime juice and zest. Adjust the icing consistency by adding a bit more cream if the icing is too thick and a bit more powdered sugar if it is too thin. You want to be able to frost the cookies without spreading too hard, but without the frosting running over the sides of the cookies! Sprinkle the tops of the frosted cookies with additional graham cracker crumbs.

Cookies will keep at room temperature, in an airtight container, for 4-5 days or in the freezer for up to 2 months.