Preheat oven to 350 degree with rack in middle. Butter 2 large baking sheets.
Whisk together flour, zest, baking soda and salt.
Beat together butter and sugar in a large bowl with an electric mixer until pale and fluffy. Add eggs 2 at a time, beating well after each addition, then beat in vanilla. Mix in flour mixture and buttermilk alternately in batches at low speed, beginning and ending with flour mixture, until smooth.
Drop level tablespoons of dough about 1 1/2 inch apart onto baking sheets. Bake, 1 sheet at a time, until cookies are puffed and edges are golden, 12 to 15 minutes per batch. Cool cookies on sheets 1 minute, then transfer cookies to racks.
Glaze cookies: Whisk together all glaze ingredients and brush onto tops of warm cookies.
Let stand until cookies are completely cooled and glazed is set.
Cookies are best the day they’re made but can be frozen, wrapped well, up to 1 month.
