Preheat the oven to 350 degrees F, positioning an oven rack in the middle of the oven. Line baking sheets with parchment paper or silpat liners or lightly coat with nonstick cooking spray.
In a small bowl, whisk together flour, zest, baking soda, and salt. Set aside.
In a large bowl, beat together the butter and sugar with an electric mixer until pale and fluffy, 1-2 minutes. Add the eggs, one at a time, beating well after each addition, then mix in the vanilla.
Mix in the flour mixture and buttermilk alternately in batches at low speed, beginning and ending with flour mixture, until smooth.
Chill the dough for 30 minutes.
Drop tablespoons of dough about 1 1/2 inches apart onto the prepared baking sheets. Bake the cookies until they are puffed and edges are golden, 12 to 13 minutes.
While the cookies are baking, whisk together the glaze ingredients. Let the cookies rest for a minute or so out of the oven and then brush the tops of the warm cookies with the glaze.
Remove the cookies to a wire rack to cool completely.