Preheat oven to 325°F. Line 2 baking sheets with parchment paper or reusable liner.
In the bowl of a stand mixer with a paddle attachment (or using a bowl with an electric mixer), beat together butter and half of the powdered sugar (100 grams / 3/4 cups) until light and creamy.
Zest lemon and measure out 1 tablespoon. Set aside the remaining zest and lemon.
Add the lemon zest to the butter mixture and beat until incorporated.
Beat in flour and salt. It will be crumbly at first, but keep beating until it comes together and cleans the sides of the bowl.
Lightly dust a clean countertop with powdered sugar. Scoop out the dough and roll into a rectangle about 1/8-inch thick. Cut into rectangles, squares, or triangles.
Transfer to the prepared baking sheets. Gather dough scraps, re-roll, and cut into cookies.
Bake for 10-11 minutes. Let cool on the baking sheets for 2 minutes, then transfer lemon shortbread to a wire rack to cool completely.
To make the icing, juice the reserved lemon. Whisk 1 to 2 tablespoons juice into the remaining 3/4 cup powdered sugar until you have a smooth, pourable icing.
Set the rack of cookies over a baking sheet to catch drips. Dip one end of the cooled cookies into the icing. Let the excess drip off into the bowl, then place back on the wire rack. Repeat with remaining cookies. Sprinkle on the reserved lemon zest before the icing hardens.
Once the icing is set, store cookies in an airtight container.