In the bowl of an electric mixer fitted with the paddle attachment, cream the butter, sugar lemon zest, and salt until it is smooth, light, and creamy.
Add the flour and beat it in until the mixture turns into a sandy crumble. Mix in the lemon juice. The mixture should come together into a uniform dough once you add the liquid.
Divide the mixture in two and flatten each piece of dough into a disk. Wrap them in plastic wrap and chill for 1 hour.
Preheat the oven to 350 ºF (175 °C).
Working with one disk of dough at a time, on a lightly floured surface, using a rolling pin, roll out the dough to between ⅛–¼ inch. Stamp out as many cookies as possible, then cut with 3-inch cookie cutters.
Transfer the cookies to a parchment-lined baking sheet, leaving a 1 inch space between them. If the cookies are too soft, chill them before baking.
Bake the cookies until the edges are just beginning to brown, about 13 to 15 minutes. Prepare the glaze while the cookies are baking.
Let cool 5 minutes then brush generously with the glaze so that the glaze melts onto the cookies. Let cool completely. The glaze should harden.
Repeat with the rest of the cookie dough, rerolling the scraps.
Lemon glaze
To make the lemon-butter glaze while the cookies are in the oven, as the glaze needs to be brushed onto the cookies while they are still warm, sift the confectioners’ sugar into a small bowl. Add the melted butter, lemon juice, and a pinch of salt. Mix with a fork until you form a smooth glaze. The glaze will thicken slightly if it sits around, so stir through a little warm water if you need to—it should be the consistency of runny honey.