Line 2 large baking sheets with parchment paper or silicone liners. Preheat the oven to 350°F.
In a medium bowl, whisk together the flour, zest, baking soda, and salt.
In the bowl of a stand mixer, beat the butter briefly, until creamy. Add the sugar, and beat until pale and fluffy. Add the egg, and beat well to mix. Add the vanilla, and beat briefly again. Mix in the flour mixture and the buttermilk in batches at low speed, beginning and ending with the flour. Scrape down the sides of the bowl with a rubber spatula as needed.
Drop the dough by level tablespoons onto the prepared baking sheets, leaving about 1 ½ inches between each cookie. Bake, 1 shet at a time, for about 11 minutes per batch. Cool the cookies on the baking sheet for 1 minute; then transfer them to a wire rack.
For the glaze, whisk together the sifted confectioners sugar, buttermilk, and vanilla. I also added in lemon zest. Brush or spoon the glaze onto the warm cookies. Allow cookies to sit on the rack until the glaze is set.