Preheat the oven to 350 degrees F and line baking sheets with parchment paper.
In a large mixing bowl with an electric mixer or in the bowl of a stand mixer with a paddle attachment, cream the butter and sugars together until combined.
Add the egg, vanilla extract, lemon juice and lemon zest.
Whisk together the flour, baking soda, and baking powder in a medium bowl until combined.
Add the flour mixture to the wet mixture and mix until cookie dough has formed. fold in poppy seeds. Be sure to scrape the sides of the bowl and be sure not to over-mix.
Use a regular ice cream scoop to form cookie dough balls (equivalent to 1/4 cup of dough or 4 tablespoons). Place on prepared cookie sheet place 2 inches apart.
Bake cookies for 9-10 minutes and let cookies cool for a few minutes, then cool completely on a wire cooling rack.
In the meantime, whisk together the powdered sugar, lemon juice, and milk in a smal bowl until smooth. Add more powdered sugar if the icing is too thin or add more milk if its too thick.
Drizzle icing on cooled cookies, let sit for few minute to set and enjoy!