Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper and set aside.
In a mixing bowl, cream together the softened butter and sugar using a stand mixer or whisk until the mixture is pale and fluffy, about 2-3 minutes. Observe the mixture turning lighter in color and increasing in volume.
Add the egg to the creamed mixture and beat until fully incorporated, about 30 seconds. The mixture should look smooth and homogeneous.
Stir in the freshly grated lime zest until evenly distributed, releasing a fragrant citrus aroma.
Gradually add the flour to the wet ingredients, mixing on low speed just until combined. The dough will come together and be slightly sticky but manageable.
On a lightly floured surface, roll out the dough to about 1/4 inch thickness. Use a cookie cutter to cut out rounds and place them on the prepared baking sheet.
Bake in the preheated oven for 10-12 minutes, or until the edges turn golden brown and the cookies are set. Transfer to a cooling rack and let cool completely.
While the cookies cool, prepare the lime frosting by mixing powdered sugar with 2-3 tablespoons of milk or lime juice until smooth and thick. Dip a fork into the frosting and drizzle over the cooled cookies, allowing the glaze to set briefly.
