Preheat the oven to 350 degrees F (175 degrees C).
Cream sugar, butter, and lime zest together in a medium mixing bowl with an electric mixer until fluffy. Add eggs, fully incorporating 1 at a time. Mix in vanilla extract, mint extract, and rum extract.
Sift flour, baking powder, and salt together in a separate bowl. Mix into the egg mixture about 1/3 at a time, making sure flour is fully incorporated before adding the next batch.
Let chill in a refrigerator to allow the lime zest to impart greater flavor into the batter, up to 1 hour. Remove and allow to return to room temperature, about 20 minutes.
Spoon tablespoons of batter onto ungreased baking sheets, leaving about 2 inches between each cookie.
Bake in the preheated oven until edges are light golden brown, about 10 minutes, turning halfway through baking time. Allow to cool on the cookie sheets.
Make sure to use mint extract, not peppermint extract.
Zested limes need not go to waste. You could improvise a glaze using the juice of the limes and a small amount of rum. Mix this by the teaspoonful into a rough cup of confectioner’s sugar until your desired consistency is achieved. I usually aim for slightly lighter than maple syrup in thickness. The result can be drizzled onto the still-warm cookies and allowed to dry as the cookie cools.
The step involving leaving the batter in the fridge to infuse the flavors is completely optional.
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