Cookies:

1. In a medium bowl rub together the sugar and lemon zest between your fingers until fragrant.
2. In the bowl of a stand mixer fitted with a paddle attachment, beat the butter and lemon sugar until light and fluffy.  Scrape down the sides of the bowl and add in the egg, yolk, lemon juice, lemon extract, and vanilla extract, beating until combined.  (The wet ingredients may look split or curdled at this point, but will come together when the dry ingredients are added.)
3. In the same medium bowl used for the sugar, whisk together the flour, cornstarch, baking soda, baking powder, salt and poppy seeds.
4. Add the flour mixture to the wet ingredients and mix on low speed, until just barely combined.  Remove the bowl from the mixer and finish mixing by hand using a rubber spatula until no streaks of flour remain.  Dough will be thick and tacky.
5. Allow the dough to chill in the fridge for 20 minutes.  While the dough is chilling, preheat the oven to 350°F.  Line two or three baking sheets with parchment paper.
6. Using a 3 tbsp sized scoop, portion the dough into 14 dough balls.  Place 6 cookies on each sheet pan.  (Keep remaining dough in the fridge until ready to bake.)
7. Bake the cookies, one sheet at a time, for 10-12 minutes, until the edges are light golden brown and the center looks pale and puffed.  Allow to cool on the baking sheet for 10 minutes before moving cookies to a wire rack to finish cooling. 

Glaze:

1. Whisk together all ingredients in a small bowl until the glaze is a thin consistency.
2. Use a pastry brush or a spoon to coat each of the cooled cookies with glaze.  Glaze will set up after about 30 minutes at room temperature.