Preheat oven to 375°F and line two baking sheets with parchment paper or silicone baking mats. In a bowl, sift together the flour, baking soda and salt. In a large bowl, using an electric mixer on medium speed, beat the butter, brown sugar and granulated sugar until smooth, about 2 minutes. Add the egg, half and half, lemon juice and lemon zest and mix on low speed until blended. Slowly add the flour mixture and mix just until incorporated.
Using a medium ice cream scoop or heaping tablespoons, drop the dough onto the prepared baking sheets, spacing the dough mounds 2 inches apart.
Bake the cookies, 1 sheet at a time, until the bottoms and edges are lightly browned about 12-15 minutes. Let the cookies cool on the baking sheets for 5 minutes, then transfer the cookies to wire racks to cool completely. Cookies are best eaten within two days.
Combine the confectioners' sugar, lemon juice, and lemon zest in a medium bowl, mixing with a whisk until smooth. Add more confectioners' sugar or lemon juice as necessary to make a thick glaze. Dip each cookie in the glaze and place on a wire rack or parchment paper.