In the bowl of your stand mixer; cream together softened cream cheese, granulated sugar, and softened butter over medium speed. Add egg, lemon juice and zest and beat until smooth.
Add the flour, baking soda, and cornstarch and mix over medium speed until all ingredients are well incorporated. The dough will be pretty sticky.
Cover bowl of cookie dough with plastic wrap and place in the fridge for 30-45 minutes to chill.
Preheat oven to 350 degrees.
Line a baking sheet with parchment paper and roll about 1.5 tbsps of dough into a ball and place on the sheet.
Bake for 9-11 minutes until a toothpick runs through the cookie and comes out clean. Transfer cookies to a cooling rack to let cool completely.
To make the glaze: Whisk together powdered sugar, lemon juice and milk until you have a nice glaze. Then drizzle the tops of the completely cooled cookies with glaze.