In a stand mixer or with a handheld electric mixer, cream together the softened butter and sugar at medium speed for 2 to 3 minutes.
Mix in the egg, vanilla bean paste or extract, and 2 tablespoons of lime juice until well combined.
In a separate bowl, whisk together the all-purpose flour, baking soda, baking powder, lime zest, and salt.
Gradually add the dry ingredient mixture to the wet ingredients in the mixing bowl. Mix until a dough forms.
Wrap the dough with plastic wrap and refrigerate it for at least one hour, or you can refrigerate it for several hours or overnight for even better results.
Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
Using a spoon, ice cream scoop, or cookie scoop, drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake in the preheated oven for about 10-12 minutes, or until the edges turn a light golden brown.
While the cookies are baking, prepare the glaze. In a small bowl, mix 1 tablespoon of lime juice with ½ cup of icing sugar (powdered sugar) until it forms a smooth, drizzle-ready glaze.
Once the cookies are done, remove them from the oven and let them cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.
To glaze the cookies, simply spoon the lime glaze over the top of each cookie and use the back of the spoon to spread it evenly.