Preheat the oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.

Combine butter and sugar in a large bowl; beat with an electric mixer until creamy. Beat in egg and vanilla extract.

Mix flour, lemon zest, and salt in a separate bowl. Pour over butter mixture; beat until combined.

Roll dough out on a lightly floured surface and cut into cookies. Arrange on the lined baking sheets.

Bake in the preheated oven until edges are set and bottoms are golden brown, 12 to 15 minutes. Allow cookies to cool, about 10 minutes.

While cookies are cooling, mix icing sugar, lemon juice, and lemon zest together in a small bowl. Spread glaze over the cookies.