Adjust the oven rack to the middle position and preheat to 350°F (180°C). Line two 12-cup muffin tins with paper or reusable baking cups; set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine sugar with lemon zest, and use your fingers to rub the mixture together until fragrant. Let it rest for 5 minutes.
In a medium bowl, whisk flour, baking powder, baking soda, salt, and poppy seeds; set aside.
Add the butter to the lemon and sugar mixture and mix on low speed until roughly incorporated. Increase speed to medium. Beat until fluffy and light, pausing to scrape down sides and bottom of the bowl halfway through, about 5 minutes.
With the mixer still running, add eggs one at a time, letting each fully incorporate before adding the next. Then, drizzle in vanilla.
Stop the mixer and scrape down the bowl. On low speed, add ½ of the flour mixture, beat until just combined, then add milk and lemon juice and beat until combined.
Add remaining flour and resume mixing on medium speed for about 3 seconds to ensure everything is well combined.
Evenly divide batter between baking cups, filling each about ⅔ of the way.
Bake until a wooden toothpick or cake tester inserted into the center comes out clean and muffins spring back when touched, about 25 minutes.
Let cool in the pan for 5 minutes, then transfer to a wire rack to cool completely, about 30 minutes.
While the muffins cool, prepare the glaze: In a small bowl, whisk lemon juice and confectioners’ sugar until smooth. For a thicker frosting, whisk in additional confectioners’ sugar, one tablespoon at a time, until the desired consistency. Using a spoon, drizzle or spread glaze over cooled muffins.