Preheat oven to 375℉. Line 2 baking sheets with parchment paper or silicone liners. Set aside.
Meanwhile, in the bowl of a stand mixer, cream together the butter and sugar until light and fluffy, about 1 minute. Add in the egg and the lemon zest and mix well, about 30 more seconds. Lastly, add in the baking powder and salt, then begin adding flour 1/2 cup at a time and mix on low speed until fully incorporated.
Using a Tablespoon-size cookie dough scoop, scoop out rounded Tablespoon-size balls of dough and place 1" apart on the baking sheets. Gently flatten each dough ball with the bottom of a greased or sugared drinking glass.
Bake for 8-10 minutes, rotating pans halfway through baking time to ensure even cooking. Cool on the baking sheet for 5 minutes, then carefully transfer to wire racks to cool completely.
For the icing glaze: In a medium bowl, whisk together the confectioners' sugar and the lemon juice until soft and glossy. Dunk or spread the glaze on the tops of the cookies and let the glaze harden before storing or serving. Cookies can be stored airtight at room temperature for up to 5 days.