Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a mixing bowl, cream together the butter and sugar until light and fluffy.
Beat in the egg, lime zest, and lime juice until well combined.
In another bowl, whisk together the flour, baking powder, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Stir in the shredded coconut.
Drop tablespoon-sized balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes, or until the edges are lightly golden. Allow cookies to cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.
For the glaze, whisk together the powdered sugar and lime juice until smooth. Drizzle over the cooled cookies.