Cream the butter. Beat the softened butter until pale and fluffy. Add the powdered sugar, vanilla extract, and lemon zest. Beat until smooth, scraping down the sides and bottom of the bowl as needed.
Make the dough. Mix in the lemon juice, salt, and coconut. Gradually add the flour and mix just until a cohesive dough forms.
Chill. Cover the dough and refrigerate for 30 minutes. Meanwhile, preheat the oven to 350°F (175°C) and line baking sheets with parchment paper.
Shape. Scoop the dough into 1-tablespoon portions and gently roll into balls. Arrange on the prepared baking sheets about 2 inches apart. Do not flatten.
Bake and cool. Bake for 14 minutes, until set and lightly golden around the bottom edges. Let the cookies rest on the baking sheet for 5 minutes, then carefully transfer to a wire rack to cool completely.
Make the glaze. Whisk the powdered sugar, lemon juice, and milk together until smooth.
Decorate. Dip the tops of the cooled cookies into the glaze, letting any excess drip off. Return to the rack and immediately sprinkle with coconut. Allow the glaze to set before stacking or storing.