Arrange racks in the center and upper thirds of the oven and preheat the oven to 350°F. Line two baking sheets with parchment paper.

In the bowl of an electric mixer, cream the butter on medium speed until it is smooth, 1 to 2 minutes. Add sugar and vanilla and beat until completely incorporated, stopping the machine to scrape down the sides of the bowl as needed, about 2 minutes.

Reduce the speed to low and add the lemon zest, rosemary, salt, flour, and cornmeal. Beat until completely incorporated, stopping the machine to scrape down the sides as needed. The mixture should be smooth and resemble soft pie dough.

Refrigerate the dough in the bowl until it is chilled and firm, about 30 minutes. Then, scoop out scant 2-tablespoon-size portions and pat them into 1/4-inch thick disks. Place them 2 inches apart on the baking sheets.

Bake until the cookies are beginning to brown around the edges, 12 to 14 minutes, switching the baking sheets from top to bottom after 6 minutes. Remove the cookies from the oven and carefully transfer to racks to cool.

Set aside the baking sheets with parchment to use later.

For the glaze: In a medium shallow bowl, whisk confectioners’ sugar with the lemon juice until the mixture is smooth.

Place the racks with the cooled cookies back on the parchment-lined baking sheets. Dip the top of each cookie into the glaze and swirl so that it covers the top, letting excess drip back into the bowl. Place the cookies, glazed side up, on the racks. Press a small rosemary sprig into the center of each cookie. Let stand until glaze is set. (Store the cookies in an airtight container at room temperature. Cookies can be prepared two days ahead.)

Tip: Be sure to use cake flour, which will give these cookies an extra tender bite, and measure the flour by sifting it into the measuring cup, then leveling with a table knife.