Preheat your oven to 350°F (175°C).
In a large mixing bowl, beat together the softened butter and sugar using an electric mixer on medium speed until light and fluffy, about 2-3 minutes.
Add the large egg and the zest of one lemon to the butter-sugar mixture. Mix until fully incorporated, scraping down the sides as needed.
In a separate bowl, whisk together the all-purpose flour, baking powder, and salt. Gradually add this dry mixture to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Use a cookie scoop or your hands to form small dough balls about 1 inch in diameter. Place on a baking sheet lined with parchment paper, spacing evenly.
Bake the cookies for 10-12 minutes, until edges are lightly golden. Centers may look slightly underbaked but will set while cooling.
While the cookies bake, whisk together powdered sugar, lemon juice, heavy cream or milk, and salt in a small bowl until smooth to make the glaze.
After baking, cool cookies on the baking sheet for a few minutes, then transfer to a cooling rack. Drizzle the lemon glaze over the warm cookies.