Preheat the oven to 350°F (175°C).

In a large bowl, cream together the softened butter, sugar, and brown sugar until light and fluffy.

Beat in the egg and vanilla extract, followed by the key lime zest.

In another bowl, whisk together the flour, baking soda, and salt.

Gradually blend the dry ingredients into the creamed mixture.

Drop spoonfuls of dough onto ungreased baking sheets, spacing them about 2 inches apart.

Bake in the preheated oven for 9 minutes, or until the edges are lightly golden.

Allow cookies to cool on the baking sheet for a few minutes before transferring them to wire racks to cool completely.

In a small bowl, mix the powdered sugar, key lime juice, milk, vanilla, and pinch of salt (if using) to make the glaze.

Drizzle the glaze over the cooled cookies, then sprinkle with additional key lime zest if desired.