Muddle the mint and lime zest together with the sugar and salt. Cream the butter with the lime sugar mixture until light and fluffy. Add the eggs, one at a time. Mix together the rum, rum extract, lime oil and lime juice in a small bowl, and then add to the mixture.
Add the flour and mix until just combined. Chill for a couple of hours until the dough is easier to handle.
Roll the dough into fat logs, and then wrap well with plastic wrap and store in the refrigerator overnight.
Preheat the oven to 350 degrees F. Line baking sheets with parchment paper.
Cut the logs into 1/4-inch-thick coins. Bake until the cookies are no longer shiny on top, about 13 minutes.
NotesCook's Note: Do not over-bake or the cookies will be tough.
This recipe was provided by a chef, restaurant or culinary professional and makes a large quantity. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.